If you are looking for something cheap and practical, go for a rigid dinghy with a rig.
At the bottom of the garden I still have a 12ft Glass sailing dinghy that we used as a tender
. It wasn't any recognised class that I know of and was quite lightly built. That's why it was pensioned off ;-))
But I have roof racked it on an Audi 100/A6, it was just about on the limit.
Sailing inflatables do exist, the Tinker Tramp at 10ft and the Traveller at 12ft. They are idiosyncratic/wierd compared to the run of the mill rubbadub and the bracing for the rig is not the best. Metzeler also made a rig for one of their inflatables. They haven't made the tinker for 15 (?) years now, so they are getting long in the tooth.
There is also a beast called a Catapult, which is a catamaran with an alli frame and inflatable hulls and a single
sail with a wishbone boom. They are fast, wet and make sure you keep your swimming in practice so perhaps not the best for a dog. They do break down and are easy to car top.
Yes try and find yourself a light rigid dinghy as big as you can handle to get it on the roof.