I have this exact problem with my 2007 460 and have not figured it out yet. It just recently started. I removed the belt and it also still persist. It happens only on a cold start and lasts for several minutes before it goes away. It does not return until the next cold start. I replaced the engine oil and filter, Then dropped the transmission pan and changed the filter just to rule that out. I've thought about changing the crank pulley but the sound does not seem to be an indication of a bad pulley from all that I've read. I don't have to rev the engine, the whine is loud with the car just idling.
The marks on the belt are only supposed to line up with the marks on the pulleys when you first put it on. Once you turn the engine the marks will no longer line up and they are not supposed. You most likely had it installed correctly the first time. I say most likely because I don't know if you lined the engine up at 50º TDC (Top Dead Center) when you removed the belt and if the engine stayed in that position during your rebuild.
You still have lots of old coolant in the engine block. In order to get it all out you will need to remove the engine drain plugs.
https://www.clublexus.com/forums/gx-2nd-gen-2010-present/884009-coolant-flush-fluids-questions-comments.html
Those offsets are not really the determining factor so they won't necessarily make the wheels poke out. You have to determine if the the wheels are low, medium or high disk which will then determine if they will even clear the front brake calipers.
What trouble codes are you getting from the car? Have you actually plugged a diagnostic trouble codes reader into the car? If so, what are the codes? You say you changed the ECM, are you talking about the computer that basically controls all the systems in the car? We need a little more info on what is ACTUALLY happening with the car.
Does the car start and then die after a while? Or is it just that the battery won't charge?
Your car had a lot of potential high dollar issues that need attention. If you can diy those issues I might say stick with it but for the money you might have to spend I say get rid of it and find another reliable car that you don't have to constantly worry about.
AC might be your biggest issue. Definitely don't drive with fuel leaking. Shops will have a field day after diagnosing all the problems. It's ok to stick with the devil you know as long as the repair is reasonable. There comes a time when you just have to move on though.
This is not the forum to buy or sell cars. I suggest you search through the vast resources that sites like Edmund, Kelly blue book, cargurus, craigslist and Facebook marketplace have to offer to find a good deal.
A transmission shop will always want to rebuild them. Unless you have some diagnostic skills your at their mercy. At least make sure they replace the rear main seal and take them pan off to check the filter as well as the fluid in the used JDM unit.