MAP Gas to anneal???

No. Way too hot...makes the final temp to time critical. Don't try it unless you are microsecond aware. I have seen the results.

No experience with a Benchsource annealer but the hotter flame as compared to Propane will shorten your anneal time. The temp difference in air is not that much (~1980 vs 2080) but enough to make a difference.
I used to use MAPP gas to cut steel instead of Acetylene. Much cheaper.