I'd say that yours is the "right" way to do it, but there is no good way. I would always recast to avoid the problem: Our car is red, my brother and I. People try all sorts of things when they have to say something like that in speech.
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Henry74Thank you for your answer enoon.All the alternatives do seem a little awkward, as you say.I'm intrigued by Our car is red, my brother and I. Can I avoid possessivness by extraposing my brother and I?Yes, that is sort of rethorical, seeing as you just did that. What I mean by that is: I don't understand its grammaticality. Shouldn't it be Our car is red, my brother