What, if anything, obligates a student to work with an advisor who lobbied for him/her to get funding?
I think it is hard to answer this question without actually knowing the system and seeing whether such switches are commonly done. Typically, admission systems are either central (e.g., US-style, you apply with the faculty and once you are admitted you get an advisor) or decentral (e.g., central European, you apply and get funded directly by a specific professor). In the former case, transfers are not entirely uncommon. In the latter case, transfers are a bit of a dicey topic as students really only apply with and get accepted by one specific professor.

Your system sounds a bit like a mixture of both - somehow, admissions are centralized, but it sounds like individual professors have so much say in the process that it in practice may well work like a decentral system. One important question may be how this "lobbying" for students works in practice. Does every professor have an "alotted" number of students that they can choose (either explicitly or by common practice)? That is, does the transferring student "take away" an alotted place for a professor and add one for another? If that is the case, I would assume that transferring will indeed lead to some hard feelings. However, if the system is more flexible, and "lobbying" for a student really just means that the advisor sent a recommendation without many more implications, I think transferring may be just fine for everybody.

In the end, I think you need to see whether transfers happen regularly, and how the students that transferred are seen by the old and new faculty. If they never seem to happen, I would assume that professors will not be thrilled about the idea.