No. The .308 piloted tap is the correct tap for the .308 pin for the shaft threads. While Brick stated he follows everything up with a 5/16" drill bit, he still used the correct tap to make the threads.
I suspect the .312 piloted tap was made to sell taps. Maybe a few people who were boring to .312 asked for a piloted tap for that bore after the .308 piloted tap for the FB pin came out. I don't know. IMO, if you are going to use the .308 FB pin, ignore the .312 piloted tap, unless you simply want another 3/8-10 tap in your arsenal for other 3/8-10 tapping where you don't need the .308 locating diameter.
There will be small variances from cuemaker to cuemaker regarding the exact numbers. Experience, methods, tooling, shaft insert type (if there is one), shaft thread treatment after tapping, etc all have an influence on the gnat's ass level. How you arrive to the gnat's ass should be decided by you over time after some experimentation and testing.
If you bore to .307, you are forcing a piloted .308 tap in. The wood will be giving way. Actually, if the bore is .30800000000 and the tap's pilot is .3080000000, that won't fit either and something has to give. The tap will win. The result is a very snug fit. If the pin is too tight when screwing the shaft on, environmental changes can cause the pin to seize. That happened to one of my local cues after the customer left the cue in the car all day during a lengthy rain. Brick won't have that problem since he chases everything afterward with .312 drill. He is in Memphis I think. Bad humidity. Maybe he has found a very snug tapping procedure and then opening the minor a little afterward is best for him.
Personally, I bore to .309. I believe a .0005 clearance on each side of the pilot will not prevent the pilot from doing its job. I live tool. I check the fit. I sometimes run a tap in after live tooling. I sometimes don't. The fit tells me whether I need to or not. I treat the threads afteward. Then I burnish the threads.
If you are going to use the .308 pin, use the .308 tap for tapping the shaft threads. Exactly how you arrive to a good bore for the tap and a good fit of the shaft on the pin should be dictated by your experience, methods, and tooling.