FWIW, to my (native AmE) "learner" ear, that word order just sounds weird, but I can't explain why it sounds weird, and I can't say whether your suggestion works or doesn't. Perhaps one of the CZ natives on the team will jump in. But "Františku, čí je ten pták? is accepted.
It sounds strange. It is possible to use "tamten" and "tamhleten" this way, but not just "ten". It is the same word order as "František, whose bird is that one?", but I have found only a single sentence ending with "...je ten?" in the whole corpus and even that one is not a good example.