Hola Natterjack /
xxx
All above statements are correct... but there are a few things to mention.
xxx
You can combine your type rating ride with the initial ATP qualification ride, provided like explained above, you have passed the written... which is easy.
xxx
However, for the type rating... you did not say which one... Fact is, a "first type rating" requires, besides the rating ride in the simulator, that you do fly the actual airplane to complete 3 landings... if that has not been changed since I left the USA...
xxx
Example... you want a type rating on a Citation CE-500, you do the entire training and ATP check ride in a simulator, then you will be required to fly the aircraft for 3 landings (airplane rental 45 minutes...?)... you first type rating, and on a CE-500, quite inexpensive...
xxx
But if you wish, as an example, a B-737 type rating, doing 45 minutes in a B-737 might be quite expensive... So, what you should do, is the CE-500 rating first (with ATP) as explained above, then do the B-737 training and rating, as you now meet the requirements of simulator only training, since you aleady have a jet type rating.
xxx
Typical prices - Citation CE-500 type rating (with ATP) about US$ 6,000, includes airplane rental for the 3 landings... and B-737 rating, all simulator US$ 7,000, but if you do not hold a jet type rating, is US$ 14 or 15,000.
xxx
Happy contrails