I do not really get what you are asking.
If you were asked to translate the Czech sentence than that is what you got.
If you were asked to translate the English sentence, then it must be "ti psi" because the subject must be in the nominative case.
If you were asked to type what you hear then "ti psi" and "ty psy" sounds different }only ti and ty differs, psi and psy sounds the same).
"the foxes are looking for those dogs" can indeed be "Ty psi hledají lišky." but it is a different sentence.
Thank you, and sorry for not being clear with my question, I hadn't considered mentioning the context the sentence came up!
I was asked to type what I heard and not aware that "ti" and "ty" actually do sound slightly different, hence my confusion - to me, they sounded the same, just like "psi" and "psy". I guess I'll have to pay more attention when listening. Thanks for answering! :)