It's a literal translation of the French idiomatic construction hommes en nous, which would typically be translated, simply, men. In French it implies a congregate body, so fellow members would also be a sensible, if not literal, translation. There is a certain sense of mutuality and group identity: amongst ourselves is part of the implication. The question in the learned member's mind is: How on earth did my guys elect this yahoo to be a part of this august body? Clearly, Dumas's election to the Académie would not be universally approved by its membership.
It's not equally clear that the Dumas in question is Dumas père, though. The Penguin Classics edition of Aurora Leigh glosses this line with a simple reference to Dumas fils, providing no details about why he is likelier meant. I believe you're right in assuming it's the father, because he was more celebrated at the time, and because the stuffy Academician's recoil at the idea fits better. The rest of this answer delves into the question: "Which Dumas did Barrett Browning mean?" without resolving it, so feel free to give it a miss.
Membership of the Académie Française was, and remains, by election. The Académie is restricted to 40 members, which means that before anybody can be admitted, a member has to die, resign, or be expelled. When a vacancy occurs, potential members can self-nominate or be nominated by any of the remaining members. Election requires an absolute majority, i.e., 20 votes (out of the 39 surviving members). This makes admission to the Académie difficult even for renowned authors. Even if Académie members are well-disposed toward an eminent author, a vacancy might never arise before that author's demise. So while being a member of the Académie is undoubtedly an honor, the inverse is not true: no conclusions about the Académie's attitudes can be drawn from a given author's not being a member. For example, Sartre, Descartes, and Molière were never members.
That said, Barrett Browning definitely makes clear that the particular Académie member Leigh encounters has Opinions™ on the suitability of Dumas as a candidate for membership. As the question says, the popularity of the elder Dumas was itself a stumbling block to his admission to the Académie, as was the prolific nature of his output. Fulminating against the fact that neither Dumas père nor Balzac (another very productive author) had been elected, the contemporary literary light Delphine de Girardin wrote:
C’est donc un inconvénient que d’être célèbre ? C’est donc un crime que d’avoir des droits ? Balzac et Alexandre Dumas écrivent quinze à dix-huit volumes par an ; on ne peut pas leur pardonner ça. Un trop fort bagage est un empêchement ; à l’Académie la consigne est la même qu’au jardin des Tuileries : on ne laisse point passer ceux qui ont de trop gros paquets...
Quote from André Maurois, "Les Dumas et l'Académie" (1956).
Translation (all translations mine):
So it's a disadvantage to be famous? So it's a crime to have rights? Balzac and Alexandre Dumas write fifteen to eighteen volumes a year; we can't forgive them that. Too much baggage is a barrier; at the Académie, the instructions are the same as in the Tuileries garden: those with large packages are not permitted to enter ... "
One sentence here captures attention: C’est donc un crime que d’avoir des droits ? / So it's a crime to have rights? What rights is de Gerardin referring to?
It’s likely that she means simply copyright, or royalty. But she might also mean human rights. Alexandre Dumas was mixed-race. His grandfather was a French nobleman who fathered a child on an Afro-Caribbean slave. That child, Dumas’s father, was freed on being taken to France from Barbados. As such, Alexandre was legally entitled to all the rights and privileges of any French citizen. His literary achievement also brought him into contact with aristocracy and even members of the royal family. Nevertheless, Dumas often encountered racial hostility. Even his phenomenal success did not shield him from overt racism:
He wore flamboyant waistcoats, green as the sea, purple cloaks, and massive golden chains. Once when Nodier saw Dumas arriving he sighed: “Ah, Dumas, my poor fellow, what a lot of baubles! Will you Negroes always be the same and forever be delighted by glass beads and corals?”.
Phillips, Mike. Black Europeans: A British Library Online Gallery feature, p.4.
So in addition to the snobbery about Dumas père's unabashedly popular oeuvre, a de facto though not de jure color bar would make at least some members of the Académie hostile to his admission.
But given the date of Aurora Leigh (1856) it is by no means certain that the Dumas being referred to here is Dumas père. By that time, the younger Dumas had made a name for himself. His most famous novel, La Dame aux Camélias, known in English as Camille, had been published in 1848. A hugely successful stage adaptation had premièred in 1852. The father was, of course, a literary lion, but the son's reputation was also well-established.
Dumas père died in 1870. Five years later, Dumas fils did win entry into the prestigious institution. One reason the younger Dumas was successful where his father had not been was his relatively puritanical outlook. Dumas père led a scandalous life. He had, apparently, 40 mistresses and fathered several illegitimate children (including his namesake). With regard to the son, Wikipedia notes:
In almost all of his writings, he emphasized the moral purpose of literature; in his play The Illegitimate Son (1858) he espoused the belief that if a man fathers an illegitimate child, then he has an obligation to legitimize the child and marry the woman (see Illegitimacy in fiction). At boarding schools, he was constantly taunted by his classmates because of his family situation. These issues profoundly influenced his thoughts, behaviour, and writing.
Maurois makes a similar observation, contrasting the father and son and explaining why it was easier for the latter to make it into the Académie:
En 1870, le vieux dieu faunesque avait « désempli » le monde. À sa place le public trouvait une noble figure, hautaine et impérieuse, qui héritait cette gloire. Le sentiment populaire rapprochait Trois Mousquetaires de la Dame aux Camélias et cet instinct était juste : les héros du fils comme ceux du père sont des redresseurs de torts. Aramis en redingote se nomme Olivier de Jalin. Mais Aramis était plus aimable. Dumas père, romantique, pardonnait tout aux passions ; Dumas fils, moraliste, en montrait les funestes effets. Les malheurs de sa mère, sa propre jeunesse, si difficile, lui avaient inspiré deux idées fixes : défense des filles honnêtes contre les coquins ; défense des hommes honnêtes contre les coquines.
Maurois, op. cit..
In 1870, the old faun god [père] had "vacated" the world. In his place the public found a noble figure [fils], haughty and imperious, who inherited this glory. Popular sentiment reconciled The Three Musketeers with Camille, and this instinct was accurate: the heroes of the son and the father are likewise righters of wrongs. Aramis in a frock coat is called Olivier de Jalin. But Aramis was more kind. Dumas père, romantic, forgave all that arose from passions; Dumas fils, a moralist, showed their disastrous effects. His mother's misfortunes and his own difficult youth had inculcated two fixed ideas in him: defending honest girls against rakes; defending honest men against vamps.
Dumas père's exclusion from the Académie was felt unjust by at least some of his contemporaries—de Girardin, for example. However, his flamboyant lifestyle, his embrace of unabashedly lowbrow writing, and his race comprised too high a bar to clear. Dumas fils, on the other hand, led a relatively austere life, worked hard to assimilate the values of the middle class, and expressed those attitudes in his work. Barrett Browning suggests, correctly, that his race and his illegitimate birth would still cause some to recoil in horror at his candidacy. But he presented a sanitized enough alternative to his father that the Académie could salve its conscience for excluding the elder by letting his glory redound onto the younger.
Newly elected Académie members take the seat of the writer whom they have replaced. Quite literally so; the chairs are numbered 1 through 40, and each member is given the specific chair of his predecessor at that position. Dumas fils, Maurois points out, exploited this situation to honor his father:
Lorsqu’on demandait à Dumas fils : « À qui succéderez-vous à l’Académie ? » il répondait : « À mon père. »
When Dumas fils was asked: "Whom did you succeed at the Académie?", he used to reply: "My father."
Maurois, op. cit.
Whichever Dumas Barrett Browning had in mind, a variety of reasons could in either case account for the raised eyebrow of the learned member she satirizes here.