We do push-ups between sets of squat (not the FSL work, usually). We force our bodies to adapt to this work and if it does make your bench go down, you need to get your body stronger and to be able to handle some push-ups.
You asked what to do and this is it. EVERYONE gets stronger and better here as we are looking at long term training, not "in the moment". Hell, me and Rhodes trained together this weekend and we did push-ups between our sets of deadlift. And then we bench pressed. So yes, I do it too.
Once your body adapts, you'll be better. And that is all that matters.