How much clearance do I actually need to leave when framing around mature trees?

Three inches is definitely too tight. For mature oaks, you want a minimum of 5 to 6 inches of clearance from the bark to your substructure, and honestly 6 to 8 inches is safer once you factor in wind sway and trunk flare. Mature oaks don't just add girth slowly over time; during a heavy thunderstorm, the entire trunk can sway and flex enought to smash violently into framing that sits only three inches away.

Pull those headers and re-frame the bays wider now while it's still just lumber and joist tape. Bumping out a box frame takes an afternoon with a saw and some structural screws. Trying to carve back composite decking around a growing tree down the road is pure misery. Trex doesn't compress or wear down gracefully like cedar—if the tree expands into the edge of a board, the bark will just lever the composite upward, snapping hidden clips and popping adjacent boards. As for the chickens poking thier heads trough, widening the gap actualy gives you better options. Right now you have an accidental pinch point. If you open the bay out to 6 or 8 inches, you can picture-frame the opening with decking cut to a clean octagon, then staple heavy-duty black hardware cloth underneath the joists to block the birds while letting debris fall through. Bite the bullet and rebuild those bays now before you start laying boards.

When you lay the decking around the trunk, do not use hidden fasteners on that inner ring. What worked for me was face-screwing those immediate border boards with color-matched composite screws instead of the standard clips.

Four years in, my oak pushed out faster on one side than expected. becasue those perimeter pieces were just face-screwed, I backed out eight screws, popped the tight boards off, buzzed another inch off with a jigsaw, and had them back down in twenty minutes. If you bury them in hidden clips, you’ll hate yourself later.

Before you worry too much about trunk clearance, have you ruled out root damage from your footings? If your framing is only three inches off the bark, I have to ask how close your piers actually ended up to the flare.

Digging post holes right there usually means chewing through major structural anchor roots. Oaks are notoriously sensitive to root disturbance and soil compaction. Have you ruled out the possibility that you stressed them enough during the dig that you'll be paying to remove dead canopy over that new composite in five years?