High cropman,
Seedlings are pretty delicate. After all, they are the babies of what is soon to be your harvest
I would say a window sill may be a bit on the cold side, but that said it depends on your windowsill

I know mine are fookin freezing but I am sure others are ok. Just put your hand there for a few minutes and feel the draft

As for putting em in your flower room, it would be fine but you would probably have two things that could happen, they could poss start bleaching. This only really happens if they are under too much light, but seeing as you lights are 5ft away this would prob not happen but not guaranteed

The other thing is your seedlings could stretch due to not having enough light
How do you win??? If you can, build yourself a little veg box, these can even be made out of cardboard if needs must,. sit your seedlings in the bottom, hang an enviro over the top of em and voila, they'll stay toasty watrm due to the cardboard insulating em

Obviously make sure the top of the box isn't covered else you'll fry your plaants.
Hope this has kinda answered you Q