The Oxford English Dictionary gives the following definition for 'Drumstick':
Used of a person. Obsolete.
The precise implication of this use in the 17th cent. is unclear, although with quot. 1602 cf. drum n.1 2b. Later examples generally have some implication of thinness or tallness.
None of the examples given are as late as the setting of the Galsworthy.
There are three other examples in the text of people being referred to as 'sticks':
His son Ernest—in the Admiralty—he thought a poor, careful stick.
Her eyes rested on Bob Pillin. “That young man's a perfect stick of goodness.”
Who would dance with a dry stick like that, all eaten up with a piety which was just sexual disappointment?
The maid at the Larne household, whom I assume is 'Ellen' is described several times as being 'little', so it may be possible that there is something of a diminutive implication in 'drumstick'. This may be supported by two of the quotations included in the OED example:
1602 T. Dekker Satiro-mastix sig. Hv Asi. Hold Capten Tucca holde... Tuc. What, dost summon a parlie my little Drum-sticke?
1836 T. C. Haliburton Clockmaker 1st Ser. xxx. 199 Every drumstick of a boy ridicules the belief of his forefathers.
Though as noted, both of these date from significantly earlier than the text in question.
There is one other possibility which strikes me, which I prefer and which might more account for the use of inverted commas for 'drumstick'. There is a reference later in the paragraph to the tea-tray carrying a liqueur bottle. A glass of the liqueur is drunk by Jock when he is supposed to be pouring it for Heythorp because he pours it into a glass which has already been used.
It isn't clear what the liqueur is, it is only described as being yellow. I wonder if the bottle or decanter is what is being referred to as 'the drumstick', some bottles do have that quality to their shape. Specifically the bottle used for Hock, Reislings Mosels etc which is known in German as 'Schlegelflasche' which translates as 'flail bottle'. A flail in this context is a mallet, which has similarity in form to the kind of drumsticks used for large bass drums of the kind used in military bands. Some sources translate the bottle name as 'drumstick bottle'.
The bottle style is also used for sweeter dessert wines which may just squeak into the definition of 'liqueur'.