Natural gas will heat with similar heating technology as propane. The real benefit is that you will not run out of fuel mid-use and you will not have to exchange containers.

Since you are already running gas, $1500 sounds really high to extend a line to your terrace. How much did they charge to stub for a gas grill? The amount should be similar with the only variant being the length of pipe.

I know about the benefits of having it permanently plumbed-in. There is also the disadvantage of being unable to move the thing anywhere (such as back-yard) or even replacing it  without hiring someone... We aren't going to use it too often  maybe, 10-20 nights a year total, I think... How much did they charge to stub for a gas grill?

We aren't setting up a gas-grill :-) If we do grill outside, I intend to use real wood, otherwise, there is a great gas-range (with grill) planned for the kitchen. We negotiated on the whole house, not per-item, so I can't tell, how much is each item.

He said, the materials will cost about $800 and the rest is labor, which does, indeed, strike me as expensive, but I can't replace him at this point without major hassle and delays (the house is half-way through inspections) and he knows it...

I recently had a new gas line put in (larger diameter) and it cost nowhere near what your being quoted. Of course, the length of the run is surely different etc. Why not get another quote if it sounds out of line. Personally, I would run a line out there given your in the construction stage. You have the option of not using and its a lot easier to extend it/move it later if you have it in the general area

"The amount should be similar with the only variant being the length of pipe."

And very possibly the size of the pipe.