I got very lucky and was able to pick up an FTW3 ultra 2080ti. This is going to be replacing my old EVGA 980ti in my current rig. I know the 2080ti is more than up to running more than one monitor but instead of banishing my old 980ti to the closet would it be a net gain in performance to run my second monitor off of it instead?

The second monitor is usually just running chrome or something simple so I can browse while I have a full screen game open on the main monitor. Both monitors are g-sync compatible if that would complicate things.

If running two different cards in the same system ends up being a configuration nightmare I’ll stick to just running with the singular new card, but if it’s relatively painless why not? Anyone have any experience with similar setups?

Depends on what you want to do. Using different model cards is only useful for desktop use and desktop use is not going to tax either a 980ti or a 2080ti. In game, Nvidia Surround requires all displays to be connected to a single card or like cards that are SLI'ed together. Mixing a 980ti and 2080ti will not work for Nviida Surround applications.
 
TLDR; Get rid of the 980ti and just do it with the 2080ti. The 980ti will not improve performance.

Got it. Thanks for the advice!