You can absolutely do constructive things in Lean, as long as you use neither the built-in "quot" type nor the three other axioms described in the docs, and then you get a system with all the desired properties I think
Some constructivists may also take offense with proof irrelevance (and the resulting loss of normalization [1] or its incompatibility with HoTT), which you can only really avoid by avoiding Prop.
[1] https://arxiv.org/abs/1911.08174