Wait- how much set did it take?
Any bow with less than 2" of set can be a candidate for sinew. I've started tillering my bows part way, about 24" or so, just as they're starting to take a little set (set on the scale, not visual). I then temper and sinew.
If the bow has taken a lot of set then the belly is going to come apart. However small amounts of set can be dealt with and shouldn't be a problem unless you're looking to really reflex the little thing.
As for how much, 6-12 tendons should be fine for a nice thin backing. Heavy backings are best for really boosting weight, and this ones can be more than enough to help improve unbraced profile.