OmniStyx ; this is what I do as well, 531 for cleans, than 531 for deads, and it's seems to work well, just hitting required reps. I don't see why the OP couldn't do somthing along these lines with press, and pushpress. That would be 5 sets each, 10 sets total, with 2 of the sets being close to failure. To me this isn't over doing it, as long as thats all the over head work you do. it's no more work than if you were doing BBB template, with either press. Me personally I would do the strict press first, but I think thats a matter of prefrance for the OP. I say give it a try, and see how it effects your strength, and numbers.
For me adding in cleans is a short term goal, to be able to clean my BW, shouldn't take long, but in your case you'll have to see if this will work for the long term, or just somthing you do in short cycles. Either way goodluck.