See bug (Comment 1 and 9, "item #2" and screenshot) for background where this was first discussed, and it was suggested I open an individual bug for this issue.
Steps to reproduce problem:
Make a 9th level human fighter with Str at least 14, give them "Weapon Training I / II" for Great Axe and Bows.
Go to inventory and give character a greataxe and a magic (+1) composite longbow (Str +2).
Look on the Character Sheet tab and notice the bonuses are incorrect.
Remove "Weapon Training(s)" and notice the base bonuses are now correct.
Add back and observe changes on Char sheet are incorrect again.
Save .pcg file and exit PCGen.
Restart PCGen, reload character and notice that now the Axe value correctly includes the Weapon Training bonus.
So, saving, exiting, and reloading seems to be a work-around. However, this bug bit me when I was first making a character, had just saved it, and went to print it out without exiting the program first.