What I'm about to say may seem a bit daunting at first but all submarine sailors do this to earn their Dolphins once assigned to their first boat.
You need to hand trace your AC distribution wiring & develop your own diagram. I've done it on my three 5th wheels to get a base line before starting...
I originally mounted my panels individually. I went with a real rail system when adding more panels. I'd go rack system from the git were I to do it again.
Remember you need to reach both the nut & bolt head when mounting. Bolting angle tabs to the panel & fastening the tabs is easier than...
I suspect there may be a time weighted parameter (week or two) that prevents notifications once it expires. I've noticed this behavior in this & other forums I frequent when I go silent for a while.
Anything below 11.5 VDC on a new lead acid battery is cause to return as defective. 12-12.4 should be what you see. I'd bring a DVM with me to verify the replacement.
What is the manufacture date code on it?
You might see an improvement using an Easy Start device. They are pricy but very effective, it can lower surge current by as much as 70%.
I'm a very satisfied customer I can start & run a 15K RV AC unit on a EU2200Si Honda in ECO mode, with other minimal loads running.
I'd suspect a current surge in phase with voltage droop. The inverter needs time to stabilize when hit with a large demand. The grid has gigawatts of available power vs a 5Kw inverter.
I'm not differing with y'all, just suggesting you ask for more data to support their recommendation. I get you are willing to accept their recommendation on face value & I'm fine with that being we've had this discussion.
I'd query them about reducing the final charge % to between 65 & 50% prior to disconnect. LiFePo4 guidance I've read would suggest 100% SOC less than ideal for storage.