Even when the higher wattage lights are on, the radio and the lower wattage bulbs wont come on. The inverter does not have the sleep feature. the inverter is a WAGAN 1500 (3500 surge) It is kind of strange to me!!!I am typing now with the lights on, and the radio won't come on, use to work.
I had a look at a picture of your inverter and noticed that it had two output receptacles. I know this is a long shot, but did you have the radio and lower wattage bulbs plugged into the same receptacle as the higher wattage lights that still work fine? And at the same time by using a multi-strip outlet? I suppose it is possible for one receptacle to work and the other to be faulty, depending on how they are wired.
A modified sine wave is an ugly thing to look at on a scope when compared to a pure sine wave, but it has little effect on most equipment. I have a 1500 watt modified sine wave inverter that I keep in my truck and I have run a 13A power saw from it on many occasions without any problems.
I have also heard of people experiencing problems with cell phone chargers on modified sine wave inverters. Your radio works on the same principle as a cell phone charger where a transformer steps down the voltage from 120V to around 20V or less and then converts that to DC voltage in the 6V to 20V range depending on the requirements of the equipment being charged. But if your radio worked before, then there is no reason for it to not continue working unless the inverter has a fault.