Basically, you have:
-main work, it's the main lift(s) of the day
-supplemental work: it's BBB, the 5x10 sets
-assistance: everything else, in the reps prescribed, and you can do them every day, each category. So every day you "should" do main work, 5x10 BBB, then 25-50 reps of each category (push, pull, single leg/core). Assistance is to be done with a grain of salt - doing pull aparts and face pulls the day before deadlifts is one thing, doing a ton of back raises the day before deadlifts is dumb.
Usually, it's recommended to do bodyweight stuff for assistance. Less stress on joints and less intensity to accumulate volume - I'd avoid weighted dips, but then again, you might be able to handle them. I honestly don't know, someone who can dip with +60kg for reps won't probably have any issue doing weighted dips with +20kg as assistance.
You shouldn't get to the point where assistance work gets you sore for days, that's why assistance should be light enough to accumulate low intensity volume without impacting on your recovery and performance.
You're already benching twice a week in BBB sets, so that's a lot of work you're already getting, which is why the extra chest work should be something like pump work if you really want to - again, it depends on you and your ability to recover, you have to base it on your previous experience and your first weeks in 5/3/1 if this kind of training is new to you.
I.e. I do deadlifts on monday, throw in about 30 reps of cable cross, then press on tuesday and do 50 reps of DB bench, then squat on thursday and throw in lateral raises for 30 reps, then bench on friday - this way alternating between chest and shoulders I don't have the problem of overlapping exercises, soreness and whatelse.
But you're more focused on chest, bench a lot and do dips, so I'd be cautious. Stick to the program for a couple of weeks at least before considering about adding more chest work if you feel it could impair your recovery/performance.
Your job is, in a few words, to do the main work the best way you can with solid reps, do the supplemental work (BBB) with sound technique and bar speed, then accumulate low intensity volume with assistance stuff.