question

Do NOT use one of those breakers you show, there are many reports on this site of people having high resistance on them melting bodies, tripping at the wrong currents etc.

You do not state what type of Multiplus you have, a 3000VA model should be fused with 400A and should have 2x50mm2 cables. I do not like that type of fuse holder with the allen key (hex key), i am not sure that you can torque it up enough. Also, the indicated hot connection does not have a washer under the bolt head, this is very important as without a washer the tightening torque is much less effective and that will make the connection contact poor. Change the bolts to proper hex head brass bolts with washers and torque them up properly.

When you mention 0.2V drop on the fuse was that measured on the bolts, cable lugs or exposed fuse blade. ab300A fuse should itself not have a high voltage drop or get hot at 150A. I am not sure you really have enough battery, again as you do not give your inverter model I can not confirm.

btw, there is nothing wrong with fuses, my 3000VA 12V Multiplus has run for years through a 400A fuse without issue as have most others.

@pwfarnell

So I replaced the Allen key bolts with hex bolts and washers and torqued them to about 10 ft lbs.

Result:

The cable lugs no longer heat up.

The ends of the fuse no longer heat up.

The very center of the fuse (where the yellow arrow is pointing) heats up to about 50 deg Celsius. Is this a cause for concern?

Voltage drop across the inverter is still about 1.1 volts when running the microwave at 150 amps.

Next steps:

Check voltage drop across the inverter cable lengths.

Any other suggestions?

@pwfarnell : I replaced the long run of cables and saw a major improvement. I also replaced the fuse and made sure every connection had proper washers and torqued the nuts myself with a torque gauge.

Thanks very much for all your suggestions that helped me understand this much better!