What size pilot hole for a 5/16"x2" lag bolt in a pressure treated pine fence post?

I'd split the difference and use a 5/32 drill....see if it goes in fairly tight. Or what I do is use a drill bit that is the same diameter /thickness as the center portion of the bolts in the threaded area. Hold the drill bit up to the threaded portion and eyeball if the drill bit is too large or small and pick one that closely matches the unthreaded portion in the center of the threads. That will give you the most bite of the threads without making it to hard to install. Rub a little bar soap to the bolt threads to make it easier to screw in if needed.

try a size bigger bolt it looks weak for a gate

Hold up the drill bit directly between your eye and the lag screw. When the shank of the drill bit appears to perfectly block out the entire root diameter of the screw, that's the right size. Just the threads of the screw will be visible projecting past the drill bit.

You can go a size smaller on the drill than that, but it won't add much to the holding power of the screw.

Treating it makes it soft.

9/64 is not right. Try 16/64 and a longer bolt. The best way to do it is drill through the post and put a bolt in with nuts and washers. Look at a cow fence gate.

Pine is soft wood. Use the smaller pilot bit.