It depends on what you are doing overall, but H2O2 will react and consume Chlorine, and MPS will not, however MPS will falsely show as CC on the DPD and FAS-DPD Chlorine test unless you use a special add on test interference remover. But before we get into that too far, let me ask how are you maintaining your pool, and why do you think you need a supplemental oxidizer? I too have an indoor pool, and rarely need supplemental oxidation, to the point that while I bought an Ozone generator for the pool a few years ago I have never bothered hooking it up, as I find that by following TFP methods I only run into a need for supplemental oxidation once or twice per year after large swim parties, and then I just use MPS.
Ike
p.s. I do note you keep your water about 5-6 degrees warmer than I keep mine