JVS

My guess is that the battery's BMS is not allowing the battery to power the inverter. So the inverter may be OK.
Can you check if there is voltage at the battery terminals?
Does the battery have State Of Charge LEDs? If so, do they show the battery as being very low?
You might just need something to bootstrap the battery into turning on.
Yes, if you can find someone who can do component level repairs. Most of the high voltage stuff is well inside the inverter, except for the AC-in and AC-out terminals, and perhaps the PV terminals if it's a high PV voltage model (you didn't say).  If it's a low PV voltage model (these are getting rare these days), then I'd say that there is a reasonable chance that nothing too expensive has blown up.
The battery would almost certainly be worth repairing, if it needs any repair at all. My wild guess is that the inverter went somewhat crazy and discharged  the battery very low. Usually the BMS would protect against it going too low, but some don't. If you can get someone  around with a highish voltage power supply (I have a dual 30 V unit that can be put in series, this would be ideal), then they can probably get the battery bootstrapped up if it's not actually damaged. Then you can see what if any faults show up on the inverter. It might only need a long press on the battery's reset button or similar to allow it to power the inverter, at least for a short time.

If it cannot then you can always just use it to control the grid input to the Axpert. 
As I am using a small 2.4kW Axpert I use a normal Sonoff just to be able to switch the grid on. I don't charge at more than 500W on the AC input to my Axpert so the Sonoff works fine. My grid input is always off. 

I think so. By choosing SBU output source priority, you are saying that you want Battery to be preferred over Utility to power the loads (Output). When the battery is full, I think it's deemed ready to take over from utility.

Indeed they have been playing around and upping the voltage due to the issues in the beginning.
This 53,8V recommendation started after they introduced the AM-2 version 2 and even that one still states 53,6V on the battery label. 
From own experience, pushing it to 53,8V makes hardly any difference and the original AM2 BMS very quickly settles at 53,6V.
The new Version 2 seems to have a more advanced BMS which does a much better top balance than the older BMS. 
In fact today is a perfect example.
Mine is set to 53,8V and you can see how the original AM2 already stopped charge and has settled at 53,6V. The new Ver 2 is still charging and will top balance for the next two hours or more to get to 53,8V 
The Hubble parallel trouble with the older and newer battery balancing they will never solve as I can clearly see the different behaviour of the BMS versions. But thats a different topic.