it would not hurt the bow to have the two backings,,,
the only disatvantage I can think of is it might add a bit of weight and reduce your cast a bit,,
but weigh the sturgeon skin, and if it is not heavy it might not hurt the performance,,,, especially if the rawhide you use is very thin as well,, like deer or thin goat,,
I have shots bows through the chrono,, and then put rawhide on with very little loss of cast if any,, nothing that using two less strands of string would not off set,,
or just a slightly lighter arrow,, etc etc etc,,
I have not used sturgeon,, but could you sand it very thin like a snake skin,,???