Too long for a comment:
I think there is a reason, why this is algorithmically solvable.
Given three side lengths $a,b$ and $c$, and a fixed number $k\in\mathbb{N}$, writing down all the (in)equalities, that prescribe $k$ non-intersecting triangles with side-length $(a,b,c)$ kissing a (fixed) triangle with the same side lengths yields a semi-algebraic set $C_k$; the configuration space of $k$ kissing triangles. For each $k$ it is decidable whether or not $C_k$ is empty (existential theory of the reals). Therefore an algorithm would start with $k=0$, increasing the $k$ until the first $k$ is found such that $C_k$ is empty and then returning $k-1$ as the kissing number. (The fact that this algorithm terminates comes from the fact that there is an upper bound on the kissing number.)
