Hey Angus, it looks good!
Quick thought. One wire runs from the controller, past the cig lighter plug to the battery, back from the battery to the cig lighter plug?
The wire to the battery carries about 2 amps.
The wire from the battery to the lighter plug carries 10 or 20A.
The 3rd wire carries nothing.
Connect all 3 wires to the lighter plug and battery.
Connect the controller wires to the lighter plug.
All 3 wires carry amps to the inverter.
Allow for fuses.
I usually do it that way, often using black & red wire from cheap #10 jumper cables.
One large wire to the battery. All the 'stuff' is in one place. May have a pic if you are interested.
Use that 12V iron. Let it get all the way hot. Solder those crimp connectors on, good and hot, lots of solder wicking up the wire inside the insulation. Give it a little flick to remove extra solder where the screws touch. It should feel solid a cm up the insulation.
Be ready for a blister from repositioning the hot yellow plastic insulation.
I would expect less problems with the inverter shutting down from a false low battery.
Less problems with corrosion, etc, also.
G-