1. Battery charger needs power. So main circuit breaker and specific microwave-battery charger circuit breaker must be ON.
If microwave display is on, then there is power to that circuit.
If no plug icon on windshield central computer console, then no signal from battery.charger.
Note: The air conditioner circuit breaker can be OFF just to turn off the noise of the cooling fan. Nothing to do with battery charger.
2. Westy coach batteries are usually 12v batteries in parallel (your Sears Platinum AGM are most likely 12v batteries). Saying "most likely" just in case I'm wrong and Sears made a 6v one which looks the same as 12v one.
Hard to see the cables in your photos, but you should be able to see how they're wired in parallel or in series.
Not to discourage a DIYer, but batteries can be dangerous and if don't have the basics, may not be worthwhile to fiddle around with this. I'm not a techie, have to read multiple explanations to see how stuff works, and need step by step paint-by-numbers instructions. Lots of stuff may make armchair sense but different if hands-on.
These batteries are heavy and hard to remove and install (drop a battery and could crack the irreplaceable water tank).
Note that some Westy owners may have two 6v batteries in series.
Don't remember for sure, but the Sears Platinum were around 100 amp-hr rated each. Fully charged, you should see 12.8 volts. These batteries can handle below freezing temperatures.
3. Auto stores can perhaps recharge the batteries (may have to fool chargers with a parallel good battery to recharge), after charged--put a load on them to see if can hold a charge, etc.
As your batteries are so depleted, you may not be able to see if anything works.
4. There was a thread on Group 31 batteries.
https://sprinter-source.com/forum/showthread.php?t=54580&highlight=Group
5. Basics
A. Battery Charger
Try doing the first step of making sure battery charger power cord is plugged into the battery charger and that the white plug is firmly inserted (it looks askew in photo).
Plug into shorepower, confirm circuit breakers are on, confirm microwave display is on, and does plug icon on windshield central computer display light up or not?
Check voltage on batteries to see if charging.
If no plug icon, then perhaps internal fuse inside battery charger.
B. Solar Charger
Figure out if solar charging system is working.
C. Connections
Battery compartment is dusty-dirty. No obvious signs of corrosion on terminals but could be a bad connection.
Fact that neither solar.charging system nor battery charger works to charge seems to suggest some disconnection somewhere.
Keep us posted.