Silly question but have you checked the alternator belt? I've seen this before on a car where it would start at night to go home from work, but not the next morning. The reason was that at night there was a heavy load on the alternator (headlights, heated rear window & fans) and the alternator couldn't charge the battery. In the morning it was light and the the loads weren't on. It charged ok. Your inverter & batteries along with the batteries could be presenting the same load. The belt did not appear to be slipping but when replaced and re-tightened it was fixed. Took me ages to get to the bottom of that one.