Two points:

1) Higgs may have been retconned into the story after the Foglios discovered he was popular. He started out a loyal Wulfenbach airman and became the secret general *after* he rescued Klaus - not in-story, but by author fiat. If that's the case nothing actually has to make sense.

2) If Higgs was already (in Foglio-brains) the secret general during the flashback, he had strong motivation to lie to the Wulfenbach troops about how badly injured he was and the source of those injuries. Telling them "Yeah I got my leg broken by a Ocnor Megaclank Mark VII while saving the Baron", when a O.M.M.VII. is known to be capable of defeating entire battalions of ordinary troops, tells them he's an incredible fighter. Contrariwise, telling them it was a swan preserves his cover. And how much rum would you have to give Higgs in order for him to lose control and start blabbing freely? Probably, an amount that would be lethal for a human (several times over).

Higgs' loyalty to the Jagers, and preserving his secret as the Hidden General, would in my opinion trump his loyalty to the Baron.