Yes it has more than one type of 2nd gen AESA since J-16's AESA and J-10C's are both part of 2nd gen and also let's include the J-11D's which is now being used for J-11B upgrades.I agree with the rest of what you said, but I didn’t mean a new AESA for Pakistani J-10Cs, just a different one. A new one would obviously cost a lot more and is unrealistic.
China doesn’t make one 2nd Gen AESA, even the J-10 from my memory doesn’t use a single type (given the radomes shape), I meant more along the lines of them changing it to make it more similar to the JFs (upscaled?) or just modified for certain things (thinking off How F7PG project went back in the day). Making it closer to JFs radar might actually save money instead, as long as it doesn’t negatively effect performance.
But even out of these three, only the J-10C's fit the J-10 and is specifically designed for it. When you say China has more than one 2nd gen for just J-10 fighter platform, I do not know what you mean. They have some slight differences in between them maybe for example the first J-10C from around 2017 with radar surely completed and tested on even J-10B models well before 2017. That particular radar might be slightly different to J-10C produced in 2021 but they should be considered the same platform. So any J-10C Pakistan buys will have a J-10C radar that has been developed already of course with changes due to operations between difference airforces.
Developing a new AESA is extremely difficult and expensive. Producing a new one may not be so difficult but comes at cost of manufacturing capacity shared between more product lines or cost of setting totally new production chain. For this reason I doubt a new AESA will be developed or produced for Pakistan. It will be some form of J-10C's current radar from 2021 just with the slight differences that are just necessary for PAF.