Not very clear on ATS (as seen from questions), but
ISTR from previous info on here that there is a short time delay in switching from shore to gen to let the gen spin up, and no 240v happens.
If the OP wires to inverter (one with no ATS of its own) instead of gen, is there any chance to get 240v briefly?
I have the shore power plug and manually plug it into either inverter, portable gen, or shore power. No need for an ATS. AFAIK they are for built- in gens, not portable gens and just make it complicated when you also have an inverter.
OP didn't say why he wants an ATS in his situation. Probably should lie down until that urge goes away? :)