Framework or tools to generate theorem prover/solver/reasoner for new logic
There are many related ways  you can mechanise your logic.


Deep embedding into one of the well-developed provers such as Isabelle/HOL, Coq or Agda. This is (almost) always possible, but makes using the embedded logic awkward.
Shallow embedding into one of the well-developed provers such as Isabelle/HOL, Coq or Agda. This is only possible when the embedded and embedding logics are sufficiently similar, but when it works it tends to work well.
Embedding into Isabelle. Note that Isabelle is a meta-tool for embedding logics, and Isabelle/HOL is but an implementation of HOL using the Isabelle tool. This is probably a lot of work, and moreover, most of the proof automation available for Isabelle/HOL is targetxing the logic HOL, so is unlikely to work for your specific logics.
Roll your own. It's quite easy to build a proof checker, should be just a few lines of code, especially if you take the LCF approach. More advanced tools are rather difficult, a modern SAT solver is probably > 10k LOC and that's for propositional logic, which is the simplest logic.
Logical frameworks. Some $\lambda$-calculi have been built particularly to enable the embedding of logics. Most well-known might be the Edinburgh Logical Framework and Pure Type Systems. Some mechanismation of those is available, but whether that is usable for you I don't know.