Cars not in your name can be taken to Laos as well with a letter of authorization. However, generally, rental car companies won't issue this letter (in rare cases they might). You can also self-drive a rental car in Laos, visit Avis. I know the general manager and founder of the company. Their cars are allowed to be driven into Thailand.
They will if it's a country, which doesn't stamp your passport anymore and there are an increasing number of those. Try Hong Kong, Singapore, Malaysia if using the autogates, even Cambodia (if arriving by air).