The answer to the question in the body of the text is "at present, no" (even for crystalline representations).

in particular, one inclusion in Kato's conjecture is known if the mod p Galois image is large. And Xin and I proved the reverse inclusion (under basically the same hypotheses).

On the other hand, for bad (additive) reduction we know almost nothing I cannot quite agree

Xin Wan and myself have a proof of the Main Conjecture for eigencuspforms of weight $k≥2$ under a few technical assumptions on the residual representation (which in particular exclude the CM case). In particular, there is no assumption on the reduction type at $p$ or on the value of $a_p$.

@FrançoisBrunault Thanks François and Keith, that's fascinating.

Salut François "and is very similar to Gauss' first proof of quadratic reciprocity" really? Isn't Gauss's first proof though Gauss's lemma, so counting multiples of $a$ in intervals? Is that really very similar?

"it is also clear that distinct injections go to distinct surjections". This seems false to me (in fact, clearly so). Take $k=1$ to see the problem.