You are not going to get a decent dual lens with remote lens camera for under $200. You need to double that budget to get anything useful.
The BlackVue DR750LW-2CH gets pretty good reviews. Its rear lens has a mounting plate. You could get double-sided cushion tape from 3M to mount it to your rear window, but if you drive a normal car with a raked rear window, performance at night will suffer due to reflected glare from streetlights.
Keep in mind that dashcams work both ways. If someone screws up and damages your car, you have video evidence of their wrongdoing. If you screw up and cause a wreck, you have video evidence of your wrongdoing and can’t get rid of it lest you be charged with destruction of evidence.
I’m not saying you shouldn’t own up to it if you screw up and hit someone, but video can be more persuasive than witness accounts, which means you might end up getting stuck for more money if there’s a lawsuit.