With nothing running the inverters still use power to keep it available, even the remote on units use a little to pop back on when they sence a need for power.

In my playing around with my small set, the power coming from the PVs unless you have a large system, cannot keep up with the demand from the inverter. This is the reason the panel voltages drop so far when the inverter is switched on and connected directly to the back of the inverter.

The readings your getting tell me the battery your using is in needs of a good long charge. when you disconnect it from the PVs it's 11.5, not good , when everything is connected your getting 12Vs, that means the PVs are doing their job. The battery will pull them down to its level. let it do that.
I'd be careful that this battery doesn't get any lower, it may not survive much longer at such a low level.
The low level from the meter readings to me is correct, BUT I would disconnect the inverter and let the panels do their job all weekend and then check them say Sunday. You should see a higher voltage reading by then, otherwise you may have a battery that has a bad cell and should be replaced.

Those 13 watt bulbs will last a long time, :-) if they weren't able to run, once on a warmed up, it still would have lasted 4 hours before dropping out.

With the panel charging the battery ( with inverter connected and idling) the 12v you see is that of a battery on charge.

If you disconnect the battery from the charging source the volts will drop, if you load it it will drop even more.

Feeding the inverter from the panel with no battery is not wise and all you are seeing is the panel volts. If you load the inverter the panel has no chance of driving it and the volts will collapse.

When it is all connected together the volts you measure on the battery are battery volts. It will be higher with the panel charging and inverter turned off with inverter loading it the volts will be lower.

You can't measure state of charge either on load or off load. You will only make sense of it with the panel disconnected or shaded and the inverter turned off and it will need several hours to loose the surface charge.

Just don't go feeding panels into inverters with no battery, if you had bigger panels you would likely kill the inverter.

I think those tiny batteries will have a desperately short life driving that big inverter near full load.

It will not be particularly efficient running a 1kW inverter at 13W the no load current may be close to the load current. The size of inverter will be dictated by the load you intend to take. If you don't want more than 13W then a 50W inverter may be better.

The final size of inverter and load you can use will be determined by the battery capability and for those little batteries your inverter is well oversize.