Also, bags can't go back through the same airport on the same leg (inbound/outbound).

So you can't route JFK-LHR-AMS-LHR.

Perhaps you could technically route JFK-LHR-AMS-LCY but I can't imagine they have a ready-made fare and if they somehow did, it would probably price up at a rate equivalent to JFK-LHR (plus LHR-AMS-LCY?) rather than JFK-LHR-AMS which is almost invariably cheaper.

(NB if your baggage allowance will sustain it and you can't short check, I'd strongly recommend that you go economy with checked luggage - especially as this now allows same-day changes, rather than one-way business). Business class might be an option but one-way business only books into the highest buckets, whereas short-haul economy fares are generally only a little over half the return (and are sometimes less than half the return when starting outside the UK).

It's up to you if you POUG but I wouldn't bother even though I've never actually flown BA economy to AMS...