They do count towards milestone rewards.

I also got the offer, but I'll have to wait to activate it to maximize the nights that get doubled as this promo is for next 2 stays. I have a bunch of 1-nighters first before a 2 & 4 nights stay. It won't get me to Diamond (will probably top out around 50ish nights for the year), but I'll manage to get lounge pass as milestone award so that's something.