Since there are only 3 "ordinates" with 3 choices each, there are only 6 possible permutations for each variable (v1 and v2). So how about a different type of form that doesn't require any checks on the selected values? Something like below. Note that my form labels are ugly. I'm sure you could come up with a nicer way to present the options.
In[26]:= perms = Permutations[{0, 1, 2}];
In[27]:= labels = Permutations[{"Most", "Least", "Excluded"}];
dir = {"x=", "y=", "z="};
opt = dir[[1]] <> #[[1]] <> " " <> dir[[2]] <> #[[2]] <> " " <> dir[[3]] <> #[[3]] & /@ labels
In[31]:= FormFunction[{FormObject[<|
"v1" -> <| "Interpreter" ->
Table[opt[[i]] -> perms[[i]], {i, Length@perms}],
"Control" -> RadioButtonBar, "Default" -> perms[[1]]|>|>],
FormObject[<|
"v2" -> <| "Interpreter" ->
Table[opt[[i]] -> perms[[i]], {i, Length@perms}],
"Control" -> RadioButtonBar, "Default" -> perms[[1]],
"Control" -> RadioButtonBar, "Default" -> 0|>|>]}, Identity ][]
Out[31]= <|"v1" -> {0, 2, 1}, "v2" -> {2, 1, 0}|>
