Winter's series wasn't with Watchers (at least I don't think so) but something like it. Meow by Kari Mouke was a man using Spike to cure his daughter (not telling lol) and Buffy being a curious kitty went along for the napping.
The description you are looking for sounds a heck of a lot like this one too:
Lonely In Your Nightmare by Bloodshedbaby Rated: R (WIP)During an alternate season 2, Buffy was kidnapped by a secret miilitary group and used for experimentation purposes. Upon her escape, Buffy must stay on the run to avoid recapture by the relentless soldiers who want her back. When she discovers she is pregnant through their experimental tampering, her desperation to stay one step ahead of them escalates. Yes, this is a vampire/slayer fic, and of COURSE it's spuffy. NEW CHAPTER ADDED (after two years, it's about time!)That story is about soldiers and Watchers coming into the (surprisingly) rescue along with the gang.
Guest:
and there were Watchers wanting to study her baby and when she's about to give birth one of them is hit with a food tray.... I think....
That part I am starting to believe is either those after the prophecy baby in Winters series or Lonely in Your Nightmare by Bloodshedbaby. Check them both out, but be prepared to read a long series with Winters. Awesome series even though it's not finished, it's enough to be well worth the read!