You need to understand the historical context for this. It all goes back to the various struggles and civil wars which took place in seventeenth-century England. Essentially the whole chapter is a "your great-grandfather killed my great-grandfather" type story.
Firstly, there's another relevant line a couple of paragraphs earlier:
Jacob Fuller, the bridegroom, is twenty-six years old, is of an old but unconsidered family which had by compulsion emigrated from Sedgemoor, and for King James’s purse’s profit, so everybody said — some maliciously — the rest merely because they believed it. The bride is nineteen and beautiful. She is intense, high-strung, romantic, immeasurably proud of her Cavalier blood, and passionate in her love for her young husband.
-- Part I, Chapter I (text available here)
Sedgemoor was the deciding battle of the Monmouth Rebellion, in which James Scott, the Duke of Monmouth attempting to overthrow and displace King James II, was roundly defeated by the king's forces. Many of Monmouth's supporters were deemed traitors, and executed or deported. This scene-setting paragraph tells us that the groom's ancestors had been Monmouth supporters who fled the country, and perhaps bribed the king in order to be able to escape unharmed.
The bride's ancestors, however, were Cavaliers, Royalist supporters in seventeenth-century England. This accounts for her father's animosity towards the groom: as the descendant of "king and country" types, who perhaps fled the country during Cromwell's time because of their loyalty to the king, he hated a descendant of men who "betrayed" the king and fled the country for that reason. We see much of this centuries-old bitterness in his choice of insults:
Among other things he said that my character was written in my face; that I was treacherous, a dissembler, a coward, and a brute without sense of pity or compassion: the ‘Sedgemoor trade-mark,’ he called it—and ‘white-sleeve badge.’
Treacherous - this corresponds to the supposed treachery of Monmouth's supporters in rising up against their king. A dissembler and coward - this corresponds to his ancestors, the Monmouth supporters who, instead of owning up to their beliefs and facing execution proudly, chose to pay off the king and flee the country to save their skins. (A "brute without sense of pity or compassion" - this seems to be the simple truth, judging from how he treated his wife.)
And so we come to Sedgemoor trademark, the real crux of the bride's father's dislike. It's not, or at least not entirely, due to the groom's own personality, but rather an age-old feud inherited from earlier generations. As for white-sleeve badge, I'm really not sure what this refers to. As Tim Lymington suggested here, white may have been a colour used by Monmouth's troops during the war. Spagirl found a source which provides further evidence for this:
Seeing his influence waning, Monmouth set out on a highly successful tour of the West Country, being received by people and gentry alike as if he were Prince of Wales. At Exeter he was greeted by 20,000 people, including 1,00 young men dressed uniformly in white, like an army offering itself to him if only he would lead.
There were also colour-coded regiments in Monmouth's army - the Red, Yellow, Green, Blue and White. Perhaps Jacob Fuller's ancestors were members of the White regiment at Sedgemoor?