This is almost certainly from using a "modified square wave" - MSW - inverter rather than a "true sine wave inverter" - the power conversion circuits in AC LED fixtures expect a sine wave input, and the way they behave when fed MSW input is upsetting your inverter.
One possible solution, short of "buy a new inverter" (a rather expensive proposition, especially for true sine) would be to feed the light circuits (only) though a small isolation transformer. The inverter power would feed into one side, and you'd feed the lights a much better approximation of a "true sine wave" from the other side. It should be a bit oversized for the load, since the MSW input will cause some heating of the transformer that a normal sine wave would not. However, the load is so small that almost any isolation transformer will be a bit oversized for it. Find the minimum size by looking at the VA (not W - power factor matters, a lot, here) rating of your lamps, and then run a fudge factor for MSW input of about twice that rating.
Unless the VA is absurdly bad for a 1.2W device, something like this for roughly $50 ($40.76 plus shipping) will probably work. You'd also need to mount it in a safe enclosure, provide circuit breakers or fuses, etc...
What watt power transformer should I be using?
You can use a power supply like the one pictured, rated for the amount of LED's you will be driving. Read the specs on the lighting. They should tell you how many watts per a given length of the strip. So if it says, for example, that needs about 1 watt per meter, then a 15-foot section would require a 5 watt power supply. Take your total length of 76 feet, and divide by the strip's requirements. It wouldn't surprise me if you ended up somewhere in the ballpark of 20 to 30 watts.
Can I run 14 gauge wire directly from a circuit 15 amp breaker to the transformer?
No. You can't put the power supply inside the wall, and you can't bring the wire out of the wall without going through a junction box. The best way to do this is to put a cord on that power supply, and plug it into an existing outlet. There is no need whatsoever to add a new circuit or connect these directly to a circuit breaker.
Can I plug in all 5 circuits of LED strips on top of each other to the output of the transformer?
Yes, you can wire multiple strings in parallel at the power supply. If the sections are short, you can also wire them end-to-end. Double-check the specs for maximum length of a single run.
Is this the recommended way of doing what I'm trying to accomplish?
The recommended way is to plug into an existing outlet. If your lighting requirements added up to 1000 watts, you'd want a new circuit. At 20, 30 or 50 watts, it is ridiculously small in household circuit requirements.
I would also like to add a on/off switch, do they make a wall switch for 12v?
Any switch can be used for 12v. I would put the switch before the power supply. Put a cord on the power supply like this cord with switch.
Can low volt wires come right out of the wall, or do they need a junction box before being connected to the LED strip lights?
The low voltage wires don't need a junction box, but they do need a junction plate, like those used for phone jacks.
Best Answer
Buy a better inverter. The DC/AC circuits in your 10W lights evidently like a pure sine wave (as provided by grid/mains power) on the AC input side, and your inverter is almost certainly not a pure sine wave inverter (just based on your symptoms, but feel free to confirm/deny).