Stas, apologies for the slow response. I was trying to think of ways that this can be done, but I don't think it is possible. UPDATE: Very possible. See my reply below.
An alternative solution would be to use separate FormFunctions. but this is not nearly as "clean". Here is a trivial example.
a2 = CloudDeploy[ FormFunction[{"x" -> {"cat", "person"}},
Switch[#x, "cat", HTTPRedirect[First@cat], "person", HTTPRedirect[First@person]] &]]
Where I have previously defined (and deployed) the secondary forms:
cat = CloudDeploy[FormFunction[{"x" -> "CatBreed"}, #x["Image"] &, "PNG"]]
and
person = CloudDeploy[FormFunction[{"person" -> "Person"}, #person["BirthDate"] &]]
