You can find a complete set of Strichartz estimates for the nonradial KG equation in Lemma 3 in <a href="https://projecteuclid.org/euclid.rmi/1049123084">this paper</a> by Machihara, Nakanishi and Ozawa. Let me add that, usually, in proofs of local or global well posedness for the wave or KG equation, the non-endpoint Strichartz estimates are more than enough. Thus the wider range of admissible indices in the radial case should not be of help, but it is difficult to say before trying of course.