Can the FAA allow a foreign training center to add a type rating to an FAA license? I have an ICAO license and an FAA one with the same type rating on both. I have done the initial training through the FAA in the US, but was able to validate the sim, training center and the DPE to get the type on my other ICAO license as well.

Now I wish to do the same, reversed - get an initial for a different type on my ICAO license and have the FAA validate it so I will have it on my FAA license as well.

Is that possible?
Training centers do not issue type ratings, examiners do. To validate foreign ratings in your FAA license you have to submit the request and supporting documentation to the FSDO closest to you. They will tell you if it is possible to add the rating to your license based on reciprocal agreements.