Adding subpanel to transfer switch
Go ahead and do that

A subpanel downstream from a transfer switch is generally an expected thing to have in a standby power system.  125A is a fine size for such (since you likely aren't going to be putting nearly that much load on the standby system), and a 24 or 30 space unit would be not at all out of place here, either.

As to neutrals? It's the transfer switch's job to deal with any neutral shenanigans that may be going on.  Since you are using your generator mainly for backup, I'd pull the neutral bond and put a clearly visible, indelible label on it saying that the bond was pulled and referring to the correct section of the manual to reinstall it if I were in your shoes, by the way -- this means you can have a solidly bonded neutral in your transfer switch, which makes life far easier when it comes to transfer switch selection.