Two things to add first, that lawnmower battery can only handle about 3 amps max charge rate without damage. It's just too small to do much. It is a starting battery and that is just about it. That said, if you want to run during the day it will work. I would connect the inverter directly to a fuse or CB that ties into the battery. If you are not protected, you are asking for trouble. Second thing, get yourself a cheep DC timer from any online retailer ($10-20). Us it and a higher power relay and use that to power the inverter for the hours of runtime you believe you can support.
That inverter will smoke that cheep PWM controller if it pulls too much power. DO NOT connect an inverter to the load side of that tiny controller. You will release the mythical blue smoke, and it never goes back in.
