My FLR also expires in early January. (I don't plan on posting off my ILR application til probably a few days before it expires, because I don't expect to have everything together until then.)
I imagine you'll be fine. You are on a settlement visa after all, which means you live here, and as long as your current visa has not yet expired you aren't doing anything wrong. If they say something, or perhaps remind you that you're soon to expire, just say you are aware of it and intend to apply for ILR as soon as possible, which is 28 days before FLR expires.
That's just my educated opinion, maybe WebyJ can give you a more definite answer.