Alas, no...I looked into getting him a provisional driving licence...but you have to send your passports *face*palm*
Ok, I think this is what I would do: Renew the passport. I had to send them additional info and because DX messed up picking up my package/ it was really hard for me to try and arrange to be home all day, the renewal ended up taking like 10 weeks. Once you have the passport, if worse comes to worse, he can apply for FLR (M) to allow him extra time to take the LIUK test... He would then be able to apply for ILR as soon as he took the test. Yes, this would be a very expensive solution to your mistake.
Who knows, if you get on this quickly, there
might just about be enough time for both, but if you don't get his passport renewed, he definitely won't be able to do either application or LIUK, and that would be bad!