Just edited this...
Well, this is my first post as a member. This is a relatively new post and I don't see any other replies. I just bought that MOES controller, and it will be arriving next week. I actually had designed and built a system to run my small off-grid system that performs this function (battery LV disconnect/reconnect/AC source switch). I have a Xantrex pure sine wave inverter and had a similar problem. The output AC on the inverter is a Ground Fault Interrupt (GFI) outlet. I believe the relays took too long to break/make (thereby tripping the GFI) and figure you might be having the same problem (or did you get it to work?). What I did was take out the GFI outlet and wire directly but added a GFI after the relay. I did look up a Renogy 1000W inverter and the one I saw did not have a GFI outlet but had an internal overload circuit of some kind (your "overload protection"). Unless you know exactly what to look for I would not recommend you getting into that inverter to "fix". If it was me (retired EE) I would open it up and have a look around to see if there's a work-around that circuit. If you do make sure you add a GFI after the relay of the MOES. Big Box stores carry stand-alone GFI devices. BTW, the reason I bought the MOES controller is the other day I went into my circuit and tried to adjust the LV disconnect level and I "broke" it. It was very much a bread boarded version with wires everywhere and I'd forgotten exactly what I'd done (about 5 years ago). I also may be heading to LiFePo battery as I'm on a trojan now with LV cutoff set to maybe 10% discharge but will be bumping that up to at least 30% discharge when I get the MOES.
One further question...I revisited the Renogy inverter and it does have a fault indicator (red) as well a ground fault indicator (yellow). I found a manual online and there seems to be about 7 fault modes for the red and 2 for the yellow. You should look at your manual to see if any of the faults make sense. Also, what are your high and low voltage settings on the MOES? I could help maybe more if you provide your exact model #.