For your system, there are others much more competent than I am, so I will leave the details and final answer to them. My first reaction, though, is that you say you shut down the solar system for a month, leaving no source of charge current. Like most RVs, a TM has a number of parasitic loads - devices which continue to draw small but non-zero current even when everything is nominally OFF. The sum of these currents will draw a Group 27 lead-acid battery to zero in less than a month. Perhaps it pulled your lithiums down pretty far, so they were hungry when you tried to charge them with the genny.
A month is 720 hours, so a discharge current of 0.2 amps amounts to 144 AH.
Were the TM shells up or down during the disconnect period?
Can you charge the lithiums, then measure the discharge current with no charging going on?
Bill
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