Eastboy

I could be wrong but it looks as if some of the diodes have blown. 
I made this mistake once a long time ago (when I was still young and stupid) by connecting a solar panel directly to a battery with no charge controller.
Maybe the customer made the same mistake. 🤔

That's lead-acid, the gel part just means the electrolyte doesn't spill out like a flooded battery would. Usually they have better cycle life than an AGM. But basic chemistry is lead-acid and the rules are similar. Same depth of discharge recommendations, can usually charge and discharge slightly higher (up to 20A no a 100Ah I'd say).

15W for the lamps, times 8 hours, that is 120Wh.
48W TV for 5 hours is, let's just say 50*5 that is 250Wh.
Total required is 370Wh. Add in inverter efficiency (85% to 90%, if we're not too optimistic) and that gets you around 460Wh. Times two, because you only want to discharge the battery to 50% (assuming you want to use a Lead Acid battery). So you need about 900Wh of battery. Divide by the volts. If we assume 12V, then you need roughly a 75Ah battery. Might as well aim for 100Ah then, those are easier to find.
To recharge 460Wh, in SA you have roughly the equivalent of 5 hours of peak sunshine, so you need to get around 100Wh into it for each of those hours, which means you need a minimum of a 100W of PV. Again efficiencies come into it. If you're using a cheaper charge controller, add 30%. Also, if you use lead acid, you really want to be finish the bulk phase around 2PM so you have plenty of time to float-charge the last few percent, so better to err on the side of bigger.
A 100Ah lead-acid battery should ideally charge no more than 15A (20A if you push it a bit), and 15*12 = 180W. So you want PV around the 150W-180W bracket, in my opinion.
Now it gets even more interesting. Again, I assumed you want to go 12V (at such low power levels that would be the way to go). For that you ideally want 36-cell solar panels, because they make around 18V at peak power. So when you use a cheaper PWM charge controller, there is less wastage. Above 150W PV modules tend to be 60-cell or 72-cell combinations, so this again cements the 150Wp panel choice. It's about the max you can go anyway, unless you upgrade to an MPPT charger so you can use higher-voltage panels.

It does, but will only work if your load is lower than the rating. 
Only if there is a correctly rated fuse between them. In your case a 125 Amp should do. 
Only if the inverter has its own low voltage cutoff protection. If you bypass the Charge controller, then all protection should come from the inverter itself. 

There is your problem. The inverter needs to be connected to your battery directly. In the photo above I see it was connected to the Output/Load of the Charge Controller. Rule of thumb, if it is a 20Amp charge controller , the load connected to it should not exceed 20 amps. Same for a 30A and so on. 

As a guess, your load is probably pulling your DC voltage down below the inverter cut-off threshold.
A 650W load at 12V is over 50Amps.
I think you start by getting a multimeter and taking some measurements.
System design-wise, you are under-paneled.
 
 

why on earth are you running a heater in summer xD
its likely the inrush current of the heater is overloading the inverter, that or your batteries are chronically under charged. Does the inverter power other loads fine?

So this is a 12V system. 
According to this you have a 1000W inverter... but the image shows 1500W printed large on the inverter. And you say it is 1200W... These varying specifications indicate a cheap brand that will lie to get you to buy their product. But sometimes they can work ok. Lets give the inverter the benefit of the doubt.  Do you know what the low voltage cutoff of the inverter is?
When trying to draw 650W from the inverter, it will be drawing about 720W (including inefficiencies) from the battery - and at 12V, that is 60 Amp.
I hope you have very thick cables running from the batteries to the inverter.  These cables must also be as short as possible, since any voltage drop over the cables is a much more pronounced loss at 12V than a typical 48V system. You can measure the DC input of the inverter with a cheap digital multimeter while drawing a load to see how that voltage drops.
Also - Why do you say "inductive load"? Heaters are normally resistive loads.  Inductive loads are normally motors, compressors, airconditioners etc. A resistive heater should be an easier load to carry than an inductive load which could have a large startup "surge" current.  
Can you upload photos of your battery wiring, and heater load so we can get a better idea of your setup?