The rules would interact only with the "first iteration." After all, if someone said "add 1 to 1" the expected response is "2" and not "How many times? Infinity? Ok, it's infinity."
Unarmed strike and natural attack damage would still be tracked separately. For example, if the character had 2d6 unarmed damage and 1d8 claw damage, having Beast Strike would just add 1d8 to the character's unarmed strike damage.
Though I'd recommend being wary of stuff like Greater Mighty Wallop. If FCT's wording is taken strictly, the size increase from GMW would also seem to apply to the selected natural weapon even if it wasn't technically an appropriate target like a claw with piercing/slashing damage instead of the GMW's required bludgeoning. Then again, one could argue that only effects that specifically call out unarmed strikes would work. *shrug*
But there's no infinite loop shenanigans here since FCT doesn't add unarmed strike damage to the chosen natural weapon. It merely allows the natural weapon to benefit from effects that would normally only apply to unarmed strikes such as Stunning Fist.