My older horse is also on Equioxx, and also needs Banamine with vaccines now. Last fall, he got his Equioxx in the am, vet appt in the afternoon with the first dose of Banamine, and then just Banamine the following day before restarting his regular Equioxx the day after.
Essentially you wouldn’t want to give them both at the same time, but my vet was okay with them spaced a bit on the day of - not something you would want to do regularly, but as a one off/occasional kind of thing it wasn’t particularly concerning. And then of course since he gets Banamine the following morning as well, he doesn’t get Equioxx that day.
I’ve also previously just made sure he didn’t get Equioxx the day of vaccines - that obviously also works, and there is probably a slightly lower risk since you’re not stacking meds that way. I have not tried using the Equioxx on its own to combat his vaccine reaction (not fever in him, more stiffness/soreness in his neck - like his neck becomes stiff as a board) so can’t comment on that.