One can't challenge an apprentice, the apprentice doesn't have any standing with respect to the Code, and there is no dispute between the magus and his former apprentice. I'd say a public challenge of this sort would do more to work against the Tremere, than for him. He would look like a bully by challenging the apprentice. Of course the full Tremere magus, with his focus in Certamen is going to win against an apprentice who may not even have any Art scores above 0, yet (there's a lot to teach besides the Arts, and opening the Arts happens first). The age doesn't say enough about how far along in the apprenticeship he is. And, the Bonisagus can always hire a champion, maybe even a champion of House Tremere, perhaps, if the House isn't behind what the Tremere is doing, this is certainly possible, and it becomes a public rebuke of the Tremere who lost his apprentice.
Taking one apprentice, as is the Bonisagus's right under the Code should have some social repercussions, but it shouldn't amount to everyone in the Tribunal being turned against them. It could involve someone who was a weak ally being neutral towards them now. Someone neutral towards the Tremere might be more favorable. A whisper campaign based on truth can be started, a whisper campaign based on lies will likely reach the ear of Redcaps, the inveterate gossips of the Order, and perhaps even a Trianomae, such as the Tenens for that particular area. These are all activities that need to develop in the fullness of time, though. It could be that the Bonisagus took the apprentice for a reason and invents a marvelous breakthrough with the apprentice's help. Won't the Tremere and all who supported him look foolish and petty for their acts against the Bonisagus? If nothing special comes of the apprentice, or the Bonisagus's work, won't he look foolish and petty, especially if he develops a habit of Boni-snatching?
Doissetep could leverage a campaign to wrest political power back from the PC covenant. Tribunal decisions could begin to be decided in the favor of Doissetep, rather than the PC covenant, because while the PC magi are busy training apprentices, or wresting on their laurels, Doissetep is now focused on rebuilding their political capital. Or something.
Simple fact is, all suggestions, including mine, are speculative, because we don't know the structure of that Tribunal in the OP's saga. He wanted suggestions that would allow a response within the Code. I've seen a lot of suggestions that could potentially be proven to be outright Code violations, especially if the Tremere and/or Doissetep isn't subtle and also patient.