It's sufficient to check it for the maps that Rezk calls "sharp" in his paper "Fibrations and homotopy colimits of simplicial sheaves." A map $f : X \to Y$ is sharp if for each base-change of f along any map into the base Y the resulting pullback square is homotopy cartesian. These sharp maps are dual to the flat maps Hopkins invented, which appear in the appendix of the Kervaire paper and which appear in other works (e.g. Batanin-Berger) being called h-cofibrations. I think of these flat maps like strong deformation retracts, and that's why I thought you might want to hear about sharp maps.

In Proposition 2.2, Rezk proves that a model category is right proper if and only if every fibration is sharp. This same result appeared in Morel Voevodsky when they were trying to prove the motivic category was right proper (it was a long proof, but this fact was crucial).

You asked if you can check it for less than the collection of all weak equivalences. In A.5 of Motivic Symmetric Spectra, Jardine proves in his setting that it suffices to check it for trivial cofibrations. It's a pretty general proof, so perhaps something similar holds for you.

I can't remember right now if it's sufficient to check for weak equivalences with fibrant codomain. I'll think about it a little later today.

You asked if you can check it for less than the collection of all weak equivalences. In A.5 of Motivic Symmetric Spectra, Jardine proves a general right properness result from Corollary A.4, which is the statement that weak equivalences with fibrant codomain are preserved by pullback along a fibration. So that appears to answer your question. His proof of Corollary A.4 uses facts specific to his setting, but it's still quite general.

Since you mentioned maps dual to strong deformation retracts, I feel like I should also advertize Rezk's work. Rezk calls a map $f : X \to Y$ "sharp" if for each base-change of f along any map into the base Y the resulting pullback square is homotopy cartesian. These sharp maps are dual to the flat maps Hopkins invented, which appear in the appendix of the Kervaire paper and which appear in other works (e.g. Batanin-Berger) being called h-cofibrations. I think of these flat maps like strong deformation retracts, and that's why I thought you might want to hear about sharp maps.

In Proposition 2.2 of "Fibrations and homotopy colimits of simplicial sheaves", Rezk proves that a model category is right proper if and only if every fibration is sharp. This same result appeared in Morel Voevodsky when they were trying to prove the motivic category was right proper (it was a long proof, but this fact was crucial). Perhaps this will also help you carry out your proof.