I've had the same problem in SoCal with my HDJ81 (1HD-T). Been looking for over 4 years and still have not found a worthy shop.
I had my full hydro A442F rebuilt by a
MC Transmission in Anaheim. They did a good job. I went through AOE, but honestly, you would probably be better off going direct.
AOE / Cruiser Tech: (Laguna Niguel) will work on these engines and i've seen them work on an HJ61 before, but they are not experts on JDM vehicles. I consider them friends and I trust them, but I am not so sure I would let them dig deep into my engine since they don't specialize in engines. If you want to go through them, shoot me a DM.
PCE Diesel: (San Diego) a phenomenal shop that did a lot of work on my buddy's 1HD-T, but are no longer taking trucks (they specialize in boat engines and Navy vehicles). I literally begged them to take mine and they said they are 9 months booked and probably wouldn't take it anyway. What was so cool about them is that they called other shops they sometimes partner with on my behalf, but they all declined to take the truck.
Auto & Diesel Service: (Santa Ana) highly skilled, but do not like working on import diesels. They had a bad experience with a Hi-Ace where the owner couldn't get parts and so the van just sat on their lot for months. They told me recently that unless the engine is easy to diagnose AND you have the parts ready to go, only then will they be interested in working on it. Worth giving them a call though, but might be a drive for you. Ask for Junior (he normally answers the phone anyway).
Top Line Performance: (Huntington Beach) these guys are great and the owner, Mario, has an HDJ81 that he just rebuilt. However, they will tell you upfront they are not experts with these diesel engines and recommended I look for a proper expert if I can find one. They will do the job though and I would put my trust in them if I wasn't willing to ship my truck off to a proper expert.
I've called various other places around SoCal and some will flat out say no, while others will "give it a shot". Obviously, you don't want to work with the latter. I called Georg at Valley Hybrids and asked for advice, as well as Dave @ Delta Vehicle Systems. Both of them recommended
Torfab (Everett, WA) and that if I wanted to really get my engine sorted, I would probably be best shipping it up to Washington state.
The only others I know that are qualified would be:
Valley Hybrids (Stockton, California - 3-4 month wait time right now),
Fit Garage (Bend, Oregon), and
South to North Land Cruisers (Garden City, Idaho - also recommended by Georg @ Valley Hybrids).
Or you can just do what i'm doing and try to learn and fix it yourself. It's expensive and infuriating, but i've made some progress and learned a hell of a lot in the process.