Offgrid L14-30 connector to solar generator
If there are no 240V loads (seems likely) and if your inverter package has an appropriate output breaker to protect itself from overload (since the panel is clearly going to be able to draw far more than the inverter supplies) you should be fine, but you're almost certainly going to need a custom-made cable, not a pre-purchased adapter. 

The L14-30 has hot1, hot2, neutral and ground. A 5-15 has hot, neutral, and ground. The expected voltage between hot1 and hot 2 is 240 volts. If there are only 120V loads, you can connect both hots on the L14-30 to the single hot on the 5-15 and it will work, up to the point that the inverter overloads (not too likely if your loads are as described, but then someone plugs in a coffeemaker or hairdryer.)

As the breakers are currently arranged, feeding only one of the two hots would leave half your circuits without power.

Most of your more serious off-grid inverters will happily supply 240VAC, and don't come with a receptacle at all, as they expect to be hard-wired. If you are going to connect a tiny inverter and keep the loads tiny, it should work with an adapter made as described. 

I strongly suspect that you can do a lot better (cost wise) with a built system than the pre-packaged one.