I prefer to cut during the growing season because it is easier on me.
Biologically, the only living part of the tree comes off with the bark. It is the cambium layer which we call the inner bark. That's the only part that has water in it besides the leaves and roots as it is the tree's transport system. That's why girding a tree kills it.
I agree with Don too. Cut some now and later. Wait a few days and draw knife the bark off. Get close with the draw knife and then take a scraper to it.
Take a stave or 2 and floor tiller them. Getting them to bend an inch or 2. Monitor the moisture level with a moisture meter. If you don't have one, weigh the stave daily and look for moisture loss to be constant for a few days.
Above all...have fun!
Jawge
If the rest of the wood didn't contain water , we wouldn't have to dry it .The Cambium produces the Phloem. The Phloem is what brings the nutrients to the new growth. The Cambium turns into sapwood, which contains a lot of water. The heartwood ,although it is essentially dead ,is still not dry. In your white wood bow ,you will be using sap wood from the outside next to the bark for your bow, so it's going to have a lot of water. (use the inner bits as well of course) There is reason behind cutting firewood in winter , taping maple trees in spring ,etc.
You can cut any time of the year, just keep in mind the water content and how it effects drying. Good luck , and have fun!