How to calculate wattage used in a solar panel set-up

Well, you aren't completely clear on your scenario, so I'll give you a number of answers:

1. How long would it take to fully charge the batteries by solar, without the "electronic" running?

Assuming everything's efficient, each of your 6V 20Ah batteries can store 120 watt-hours, for a total of 240 watt-hours. Your solar panel puts out 200W, so it would take about 72 minutes to fully charge the batteries.

2. How long could the fully-charged batteries drive the "electronic", with no solar cells?

240 watt-hours, drawn at a rate of 300 watts, would last 48 minutes.

3. How long could the fully-charged batteries drive the "electronic", with the solar cells supplying power as well?

The device draws 100W more than the panels can supply, so the batteries would last 144 minutes.

Note that this all assumes you're getting a full 200W out of your panels, which will only happen in full sun, and only when your panels are facing directly at the sun. So, depending on your circumstances (weather, panel placement, trees, etc) you're going to have to cut the solar panel power by a significant amount.

I’ve not seen a golf cart battery that was only 20AH , the 215 number sounds plausible so you would have 215 Ah at 12v with 2 batteries.

Are you sure you have your numbers correct? My guess would be you have an inverter that outputs 300w at 120, to do this it would draw ~25 amps from the batteries (the draw would be more because of losses in converting 12vdc to 120v ac).

If you want your batteries to last don’t discharge past 50% but golf cart batteries “deep cycle “ can go down to 80% but will only last a couple of years. Using 50% this would provide around 4.3 hours of run time from the batteries alone 25amps or 107.5 ah used , your run time would be extended by adding the solar if in full sun your total run time could be 11.8 hours as you would only be drawing ~9 amps from the batteries but you know your not going to have full sun and this would leave the batteries in a depleted condition or at 50%. If you provide more information on your inverter we may be able to provide better examples.