the pd4045 and the pd5000
I installed the PD5000 for one purpose: take AC current from the PD1200 to power the AC outlets when shore power is not available. The AC side of the PD4045 requires shore power. I could have wired all my 12-volt lights to the PD4045, but I spread them between the two just for easier management. The PD1200 also routes power to the outlets when shore power is available. The PD1200 is intelligent enough to know from which source to draw its power.
Okay my first guess in the battery chemistry.
The PD4045 has a switch labeled LA/LI (Lead Acid/Lithium). It is set for LI. According to Progressive Dynamics that is the correct setting for a Lithium battery.
it looks like the pd4045 can put out 45 amps at low voltage, so its probably sending all that power to the battery to charge it. Did you change the charge profile to lithium? and you may need a 45 amp fuse.
Fuse size: The first fuse was 40 amp. It melted before it finally blew. I've been trouble-shooting with a 30 amp fuse. Removing one device at a time (PD1200, solar controller), then checking the fuse. It starts to heat up after about 5 minutes.
Everything works when on battery power, so your first paragraph is on the right track. When DC power is flowing from the battery, everything is cool - literally. When DC power goes the other way, it's not cool.
I hope that helps. Any other ideas will be appreciated.