Your batteries should operate the slide. I've operated my slide without any batteries on the trailer and just my pickup hooked up and running through the 7 pin trailer plug. As far as generators, I'm a fan of the champion 3100 inverter generator, either in manual or remote start models. It runs my 13.5k BTU AC no problem when we boondock in the boondocks. It's also quiet, has wheels and a handle, and is an inverter so it's nice clean power. I chain mine to the bumper or tongue of trailer and roll it underneath when not in use.
Used it for 12 hours straight running the trailer and satellite dish with TV on one camping trip without a hiccup. And they are cheap, around 800 for the non remote start unit. You can order them on amazon and get free prime shipping if you are a prime member and not pay tax. It's also got a fuel shutoff switch (t the honda does not), so you can throw the switch and just wait for it to die, guaranteeing no fuel left to sit in the carburetor. Keep it fueled with ethanol free gas and treat every tank with stabil, change oil once a year with quality full synthetic and get years of trouble free use out of it.