Hello and welcome the the forum. As with any scooter or motorcycle, sitting or operation in rain is not recommended. Regardless of what brand you purchase, a good quality cover would be highly suggested. As for the Like, as the other post mentioned, the main thing going for you here is the fuel injection. I can't stress how great of a feature this is. Like 200 or not, I would not consider any ride without fuel injection for the type of riding you mention.
As for the build quality of the Like, again the first post nailed it, don't expect any water tight seals in the front storage. The truck and under seat areas seem pretty good. As someone who has seen the "insides" of the scoot first hand, I can say the quality of the wiring, connectors, and various other components are as to be expected...which is very good but not quite to pair with the Honda/Vespa stuff. The fuel lines and various vacuum tubes and even the battery are of very good quality . You may want to upgrade the tires for example. They are good Kendas, but the stuff Dunlap is making are way better in the wet. I don't think you'd have any issues with the Like as long as you maintain the scoot and use a cover to minimize the effects of inclement weather.
You are getting a great scooter for under $3,000. It is what it is, but common sense and careful maintenance should get you a long lasting scoot.
Hope this helps, best of luck in whatever you end up with and as always, be safe out there.