Can I define a unitary representation of the Lorentz group on the Hilbert space of a theory which breaks Lorentz invariance?

If I understand the question correctly, then your question is more general than relativity. For example, you can ask the same question about rotations in a non-relativistic theory. In the spirit of trying to address the problem in the simplest situation possible, allow me to even downgrade to regular particle quantum mechanics, not a QFT. The Hamiltonian is of course, $$H = \frac{\textbf{p}^2}{2m} + V(\textbf{x}).$$ Let's look at the rotation $R_{\hat{n}}(\theta)$, which is the rotation around the unit vector $\hat{n}$ by an angle $\theta$. The action of rotations on the Hilbert space is defined regardless of what is the Hamiltonian. For example we might simply define on the transformation operation on the Hilbert space as, $$U(R_{\hat{n}}(\theta))|\textbf{x}\rangle = |R_{\hat{n}}(\theta) \ \textbf{x}\rangle.$$ This is true whether the rotation is a symmetry of the Hamiltonian or not.

Whenever you meet a new operation, you need to figure out how it acts on the Hilbert space. Even though all symmetries of the Hamiltonian are unitary, not all unitary operations are symmetry of the Hamiltonian/Lagrangian. I hope this helps.