it would actually be "built-in" and I suppose you could use it to describe a restaurant. I hear the term "built-in" most often in relation to houses and cars. "Our house has a built-in fireplace." "My car has a built-in phone charger." "My home has a built-in restaurant." Generally it is used to describe things that you would not necessarily expect to be included, but I don't see a huge problem with saying the hotel has a built-in restaurant. As for your "reservation for a room at a hotel", it doesn't sound awkward at all to me. I am sure I have actually said this before on multiple occasions. It's not redundant by specifying that you are reserving a room in a hotel because there are other establishments that you could reserve a room in (bed and breakfast, motel, etc.). Hope this helps :)

"With an on-site restaurant" or "with a restaurant on site" is definitely the way to go.

'Built-in' wouldn't be appropriate here. You could say 'integral restaurant' or an 'on-site restaurant'. 'I want to reserve a room in a hotel that has an on-site restaurant.' sounds fine. There's no problem with the word 'reserve' or 'book'. I've changed 'with' to 'that has' to avoid the ambiguity of 'a room with a built-in restaurant'. Your original sentence could be interpreted as your wanting to have a tiny little restaurant built into the corner of your hotel room! I know it's a silly idea, but that was what it seemed to suggest!