Technically we already do, the Ki-76 and Ka-1. The IJA didn't have any arrestor cables on their CVLs/CVEs, so they had to just use planes that could take off and land on a tiny runway on their own.
Then again, the real-life J7W Shinden was purely land-based, and Kancolle includes a fictional carrier-based version as the Shinden Kai. So maybe at some point Akitsu Maru or some other not yet implemented IJA ship will get a Kai Ni that allows it to use Army fighters like the Ki-61 or Ki-100.