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You absolutely can use clmul for Galois Field arithmetic, of course, that's why it was introduced. You can do the modulo reduction in two clmul, e.g., see https://arxiv.org/abs/1503.03465
Forgive my ignorance and maybe it's in your paper and I just don't see it, but it's not obvious at all to me how to use that to implement GF(p) arithmetic (not GF(2^n) but a non-2 prime field). Nils: 45*299 + 1 = 13456 = 2^4 * 29^2 afaict. So no such GF exists?
Nils: Ah! It's 45*2^29+1. 
Yeah, that's a prime field, everything alright there then.
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