On a generator with a GFCI, the neutral and ground are tied together PRIOR to the GFCI on the generator chassis (those I have seen).
So what happens with a GFCI is that it measures the electricity going out one current carrying conductor and EXPECTS to see the same returning on the other current carrying conductor.
But if you connect the ground from the generator to ground at the electric panel (bonded to neutral at the main panel as it should be), then some electricity will travel back to the generator via the ground wire to the generator and this will trip the generator GFCI (as it notices not all of the electricity is returning on the other current carrying wire).
Briggs and Stratton has a transfer switch which switches the neutral "off" from the main panel and connects it to the transfer switch panel neutral at the same time. So you could call this a "switched neutral transfer switch". This solves the problem of a generator with a GFCI. Basically you wind up with a "subpanel situation" where the ground is not bonded to neutral at the panel which is being powered by the generator and this all is disconnected from the main panel which does bond ground to neutral. Then the "main panel" (where neutral is bonded to ground) becomes the generator prior to the GFCI.
I don't know if Briggs has an automatic switch or not, might want to call and ask them.
Here is technical info on the Briggs switch...
Here is a wiring diagram of a generator with a GFCI...
(Note chassis ground on neutral prior to GFCI)
Briggs and Stratton web site...