Hi Elizabeth. You really do not need that powerful an inverter and it will strain the charging system on your car and run down the battery. I would look for something in the 100-200 watt level. Do you want the flexibility and portability of a cigarette lighter adapter or a hard-wired, permanent installation? The general rule of thumb is that loads exceeding 150 watts will require a direct connection to the battery. A 150 watt inverter will let you power laptop computers, small power tools, most small appliances, and even a small (13") TV.
A high quality 150W inverter will handle that load easily. Now you say you use a splitter. What else is connected to the splitter? I would avoid using a splitter and put everything on the inverter. If you want, you can try a larger inverter since if it overheats it is just going to blow the cigarette lighter fuse which is 15 amps and no permanent damage will be done. Make sure you buy an inverter with a cooling fan if its larger than 150 watts.
Hope that helps in your decision! Good luck!