Solving atomic chess is probably only feasible if it is a white win, like loser chess is.
To try to solve atomic chess I would probably write/acquire a strong atomic engine first.
Then you build a tree starting from the starting position. If it's white to move you chose only one move, if it is black to move you try all possible moves. The engine evalution will guide you in which white move to chose and which black moves should be investigated deepest.
Basically I would try to build a tree that has leaves with a certain minimum positive evaluation. This way you always have a form of very weak solution of atomic chess, i.e. a tree that tells you how to play with white to get an advantage of at least +x.yz.
You'll try to improve x.yz by analysing those parts of the tree where the leaves have the lowest evaluation. If x.yz gets high enough you can try to work positions out until mate (probably you'll need some tablebases at that point).
If you start this kind of project you probably quickly get an impression, whether getting a significant white advantage is generally easy or difficult, which might be a pointer to whether it is ultimately solvable or not.
This is pretty similar to building a repertoire, so I would probably quick start the tree with an existing atomic chess repertoire.
Of course odds are that your tree will never converge towards a strong solution, but in that case at least you've created a really nice atomic repertoire.