Does the 3 pole switch serve 3 wire load (no neutral) or 4-wire (solid neutral) load?
If 3P ATS is serving 3 wire load, you must ground the gen N as the system will be ungrounded while on the gen. The 4P ATS will disconncet the N to N conncetion between the gen system and the utiltiy (assuming the other source is utiltiy).
If the 3P ATS has a solid neutral, adequately sized for the full load service of the generator capacity (this is very important) AND is connceted to the the utility system N which is effectively grounded, you need not ground the generator, in fact you should not as you can not ground the utitliy N down stream of the main device per NEC.
If your 3P ATS has solid neutral but it is not sized for the generator's full load rating (equivalent service size) then you need to upgrade that N.
Refer to NEC article 250 and IEEE Green Book for reference.
Having GFP adds to the complication but not having it does not mean that you can ground the neutral multiplie times (other than at the main device of the source)
You may also choose to "jumper" the N terminals in the 4P ATS and treat the system as not-separately derived system and not ground the generator. Still each of the solid N must be sized for the service equivalent to generator rating.
Too much to digest?..review again...