You're right - the gun is pretty cheap. I have a Harbor Freight benchtop blast cabinet, and it was also pretty cheap - but it did a sufficient job for a long time. After blasting several FN-FAL builds and a few AR lowers, I rebuilt it using some higher quality valves, etc. I need to scrap it or rebuild it again now.
That little handheld blaster will take awhile to get the job done, but if you're patient, I think you could make it work. Keep in mind that since it's being used out in the open, it will create a lot of dust. One possible way to prevent that is to build a makeshift blast cabinet out of a big cardboard box with a clear plastic sheet over the front (even just clear Visqueen). Then, cut a hole and stick a shop-vac nozzle in the box to suck up and contain the dust. Otherwise, you'll have grit over everything in the basement, including your wife's stuff. Which would cut short your gunsmithing activities, I'm sure!
Your little pancake compressor will be slow, but will accomplish what you need eventually. I have a large, single-stage 40 gallon Craftsman compressor, and I still have to wait for it to catch up.
Regarding the grit, I've been using a blend of glass beads with aluminum oxide - thought I got if from Harbor Freight, maybe not. I don't recall the exact grade, but I think it's around 120-180. The blend seems to work pretty well in keeping the alox from getting too aggressive and rounding off engraving or sharp edges. I've been happy with the result - it leaves a nice matte/satin finish that takes anodizing well and hides any minor surface blemishes. Hopefully you'll get some other recommendations from other members.