Is the following phrase (grammatically) correct?
Bill Gates, whose company is very rich, is famous.
(I couldn't come up with a better context-free sample)
I didn't even see the second "is" as a problem. In my mind, the jump to the subsentence using commas is very orthogonal to the rest of the text. The "whose" is correctly used but if you're free to rephrase, you might consider the following, too.
It's a bit unusual to hear that (at least amongst people I talk to) but I love the clarity of what's pointing to what. As this, a bit digressive, example.
It's a true story, too, as of two weeks ago. Poor bike, the owner of which is me, is at the shop now.