Bootcamp workout

For a change of pace, I’ve been doing a “bootcamp”-type workout: 6 am running up stairs, step-ups, some sprinting and jogging, lots of push-ups, abs, a bit of high-rep (e.g. 30) dumbbell work for arms, etc.
Mon/Fri: leg-heavy, T/Th: upper body; Wed: running.

I’m cutting for a few weeks and I’m wondering how to fit in some heavy weights to keep from losing too much strength/size. I’m thinking one or two days a week of 5x5 (e.g. bench, horizontal pull, and pull-ups, plus some legs on another day).

My question is mainly, does this make sense? For example, I pretty much have to do the bench press either the day before, of or after I do like 80 pushups (to failure, eventually), when I’m maybe still a bit sore.