For one the threads are different. Those aren’t NPT. Plus, look at the bottom of that unloader. It has some sort of a valve or something on it.
I use to have an old Karcher hot water washer that my father-in-law gave me. It hadn’t been used in years. Anyways, it had an unloader built-in to the top of the pump. I was able to bypass the unloader but drilling and tapping some small passageways and installing a small socket plug. Rex at PWP told me how to do it. That guy knows every pump inside and out. The pump I had was like 25 years old. I just told him the model number of the washer which was very rare and he knew exactly what pump it had. I was in the same position as you in that I needed that pump because of the way it was setup to turn the blower on the burner.
Here’s what I would try. I would fix your unloader. You can get custom made springs made if need be but it’s like a $35 minimum order. I would then max it out and use a second unloader. Sounds like a few have done that with success. Another option is a new belt drive pump. Get like a 4 belt pulley so you have two for your engine, one for your burner, and one for the alternator. It’ll take some fab work though. That was my next plan for mine but I got sick of messing with it and finally just bought a new hot water machine. I still have the old. I might try and get it going for my buddy and his farm equipment.