Wouldnt that add to what can go wrong?(APU problems)..
You are two pilots with 2 gens,so why not let the PF fly while the PNF start the APU as you well did in the event you described earlier?.
Is your airline ok with this personal procedure?(decreased APU life/increased cost/fuel..)
APU for bleeds off t/o is the normal procedure for all 737s.
I agree with the answer of Post #3