there are lots of complaints here re amex travel and amex plat/cent travel claiming the nightly rate factors free night in, even if it doesnt, or possibly inflating the rate to negate the free night

manager at FS (or FS corporate) may be able to at least give you information

fourseasons.com/kyoto/offers/stay-longer-third-night-free/
not currently listed as a US FHR offer
amex travel sometimes has offers