If we define $\mathcal{T}=i\sigma_y K$ where $K$ is complex conjugation, i.e.
$\mathcal{T}\psi_\uparrow \mathcal{T}^{-1}=\psi_\downarrow, \mathcal{T}\psi_\downarrow \mathcal{T}^{-1}=-\psi_\uparrow$,
Then naively a term like $\Delta e^{i\Phi}\psi_{\uparrow}^\dagger \psi_{\downarrow}^\dagger$ is not invariant under $\mathcal{T}$. This is basically the problem you encountered, phrased a little differently. However, it does not mean that the system actually breaks $T$, as all physical observables will be invariant under $\mathcal{T}$. The resolution is in the definition of the transformation of $\psi$ under $\mathcal{T}$. Let us modify the definition to be
$\mathcal{T}\psi_\uparrow \mathcal{T}^{-1}=\psi_\downarrow e^{-i\Phi}, \mathcal{T}\psi_\downarrow \mathcal{T}^{-1}=-\psi_\uparrow e^{-i\Phi}$.
This phase is not observable (it is essentially redefining the phases of the basis states of the second-quantized Fock space, which has no consequences on physical observables), so we are free to do so. Notice that the important algebraic relation $\mathcal{T}^2=-1$ (for the classification of TI/TSC, etc.) is not affected. Then the pairing term is invariant.
Of course, this only works when $\Phi$ does not depend on positions. Otherwise (e.g. when there is a vortex) one can not get rid of the phase, since $\nabla\Phi$ is an observable, the supercurrent.
Best Answer
Obviously if one starts from spin-1/2 physical electrons and want to get effective spinless fermions, one has to break time-reversal symmetry.
But let us imagine that we just have spinless electrons to begin with, and time-reversal symmetry is just complex conjugation(since now electrons do not have any internal degrees of freedom). Assume that $\xi_k=-2t\cos k-\mu$, corresponding to a uniform nearest-neighbor hopping with strength $t$ and a chemical potential $\mu$. Then the Hamiltonian as written is time-reversal invariant and there is nothing wrong about it. So if one keeps this definition of time-reversal symmetry, the Hamiltonian belongs to the BDI class. However, one can add time-reversal breaking terms (the simplest thing is probably adding some imaginary next-nearest-neighbor hopping) to the Hamiltonian. The universal physics of the Hamiltonian (e.g. Majorana zero modes on the edge) is completely unaffected by the addition of such terms, provided one does not close the excitation gap. Then the Hamiltonian belongs to class D.