You already understand that you need the owner's (rental company's) permission to cross the border, regardless of what the respective governments might say. If you can't get that permission, everything else becomes irrelevant. I'm doubtful there is a workaround, but I'd sure be interested in hearing if you find one.
IMO, an expectation of finding a reliable agency on the internet charging US$10 per day for two weeks is pretty unrealistic. You might find a little Chinese scooter of some sort for that price, most likely by asking at your hotel once on the ground--quality varies greatly, and I'd be reluctant commit without seeing an actual bike first. Finding a Honda 250 for $10/day would be unlikely to the point of absurdity.
I've rented Honda 250s in Laos and Vietnam, but I assume you're looking for post-COVID information. Cheaper rates were available everywhere for cheaper bikes.
Hope that's helpful.
Mark
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