What is limit on performance of edge-preserving filters for low signal-to-noise regimes?
This is Gibb's phenomenon (just not for the common edge case).

Hence, with a linear filter, the better you remove noise (which is, low-pass filtering), the more you remove the edges (which inherently have energy everywhere in the spectrum).

There's no minimum case – the noise bandwidth you allow is always the same as the signal bandwidth you allow, and thus, complementary, the amount of signal energy (and hence, edge precision) you cut off.

Among the things you can do is apply a clever signal model. Does your signal always look like your $x(t)$? Great, so you only need to estimate one parameter ($\alpha$) to fully describe it!