Being an avid turkey hunter myself, I see both sides of this issue. One thing to consider is the size of the plots. If it's a large plot, it could be very useful to a nesting hen. If your plots are on the small side, chances of nesting success in them is much less. Think of it this way, would it be easier for a predator to find a nest or poults in a half acre field or a 5 acre field? Small plots are often killing zones for predators just like they are for hunters.
If your turkey hunting members are knowledgeable about this they may not be opposed to planting some of the smaller plots now and the larger plots later. Secondly, you could always call your district biologist and get an educated opinion on the matter and present it to the club for discussion.
He said they don't want to bushog until after turkey season, not nesting season. Those poults aren't going to hatch until well after turkey season.
If it's the turkey hunting they're concerned about, shorter grass in field improves the hunting in my experience. If they're worried about poults you shouldn't cut the fields til late summer.
You'll be fine planting after turkey season, but if the rest of the club says says it isn't enough time, tell the turkey hunters that you'll plant some chufa plots if they let you bushog now.