i would rough out, ,to floor tiller put the sinew on ,, and let them cure together,,
as stated coat the back,, or it will check,,
if you want the bow to cure faster,, take off the sapwood now and rough it out,, I would put a good layer of hide glue on the back till you get ready to sinew,,that will help with any checking, that being said, if you gonna sinew back a little checking will not effect your chance of success,,
at least rough out and put one layer of sinew on,, you can always add if you need,, I am not big on waiting
if you want a really cured stave then get one and do another bow,, and see how they compare,,,
a month will not do much in the current state,, maybe a year,,