If I were in your situation, I would take what I could get from the electrician, and then do the job right myself. Trying to fight the electrician and inspector isn't worth your time, effort, high blood pressure, etc.
The first thing I would check is that if I could make what is currently installed work. Specifically, if 2-2-2-4 aluminum rated at 90A will be sufficient for your shop. You stated some of your intended future electrical loads. Here's what I would consider as a possible issue, running a plasma cutter. IF you are running a plasma cutter, that also means your 5HP air compressor will be running simultaneously. Guessing 20A for the plasma and 30A for the air compressor, that's 50A total. Lights could be added in as well, but fairly small. Outlets are insignificant. The big unknown for me is how much current your ELECTRIC HEAT will require.
Let's say that you can get your maximum, worst case current down to 90A. That means you can use the 2-2-2-4 that is already buried and it will meet code for 90A, but still have a voltage drop issue when running the near 90A.
Next, I would verify that your 2-2-2-4 is really URD, or is actually "mobile home feeder". I don't remember seeing in your pics whether the writing on the cable was URD... You are hoping it's multi rated, specifically rated USE, RHH, RHW. If it's NOT rated USE, they by code it's not allowed to enter your house (or the shop). It has to do with the type of insulation on the wire. If the wire is URD, then you have to have junction boxes on the outside of your house and shop and transition to approved wire types to enter the structures.
If it's USE, then I would simply remove the #4 copper between the junction box and your main panel and replace it will #2 aluminum (rated USE THHN, THWN, etc.). Then you'll have your entire run rated for 90A. Then swap your breaker out for a 90A breaker.
I suggested using #2 AL instead of copper to prevent having to connect copper and aluminum together. The blue "smurf" tube meets code and I would probably leave it there as long as that area isn't subject to potential damage from people leaning equipment against it. I would replace those smurf tube connectors though, it looks like the electrician used PVC connectors instead of those specifically designed for the tube.
Norcal and wyliesdiesels have given you good advice and they know their stuff. If you've perused this group much, you will come to that conclusion.
Many licensed electricians, and inspectors wrongly "interpret" the NEC and consider all subpanels as "services" to a "dwelling". Thus lots of non service installs end up with 2awg aluminum, on a 100A breaker.
If you can do what I suggested, everything will be to code, and you shouldn't notice the voltage drop except when running multiple large loads simultaneously. Do you know how much current your electric heat would take?
Brian