I find it hard to believe that a proof or disproof of any mathematical statement could cause "many careers in tatters".
If professional mathematicians around the world all made the same mistake, it hardly reflects poorly on any individual mathematician. Moreover, when a mathematician makes a significant mistake, it almost always is for some non-trivial reason, and the disproof is highly interesting. E.g. Euler's conjecture relating to orthogonal Latin squares (Wikipedia) is still studied hundreds of years later, despite it being wrong in every single case Euler did not prove himself (i.e., he couldn't have been more wrong).
If someone disproves e.g. the Riemann Hypothesis, the likely result would undoubtedly be intense mathematical interest in the Riemann Hypothesis. Mathematicians studying the Riemann Hypothesis would likely have a massive boost to their careers as people update their theorems with the new knowledge.
If you want a plausible scenario where a disproof of a theorem could result on some egg on mathematician's faces, I suggest the Classification of finite simple groups. This theorem is proved, but the proof is so long that it's plausible it contains an error somewhere. Again though, it's not going to destroy anyone's career.
But I think there's a better idea...
Automated theorem proving
If you want mathematicians to "choose between the truth or their prestige", I recommend looking into automated theorem proving, i.e., computers automatically generating proofs. In fact, the computer can be used to make conjectures, and subsequently prove them too.
Mathematicians instinctively hate "proofs by computer" and consider them inferior because they don't give human intuition. Thus it's within the realm of plausibility that mathematicians don't want to accept the truth. As proof of concept, see for example the drama surrounding the computer proof of the Four Color Theorem.
This result was finally obtained by Appel and Haken (1977), who constructed a computer-assisted proof that four colors were sufficient. However, because part of the proof consisted of an exhaustive analysis of many discrete cases by a computer, some mathematicians do not accept it. (Mathworld)
It is within reason that "what a mathematician does" changes radically because of some brilliant ideas in the area of automated theorem proving: many mathematicians are quickly rendered obsolete, replaced by automation (like factory workers previously).