Can I play Borderlands 2 GOTY with a vanilla disc after installing DLC?

Yes, you can. The GOTY disc is the same as the vanilla disc, but also includes the DLC. Once the DLC has been activated on your account and installed on your console, you can use the vanilla disc or the GOTY disc interchangeably.

I can't confirm this will work from experimentation with those exact discs, but I can be pretty confident that everything should be fine.

Given that the DLC is designed to work with the base game, it would be extremely strange for a different version of the base game to be shipped with the DLC.

I've had plenty of other games that came in a 2-disc set with DLC on the second disc, and those were all completely compatible with the original disc.

If they weren't compatible, it's doubtful that the game would have launched at all.