How much does your SDK do? If it’s just wrapping calls to an HTTP API, use something like OpenAPI / Swagger to document the API, then auto-generate client libraries based on the OpenAPI specs.

Then if you add any language-specific niceties on top of the auto-generated code (i.e. accessor functions to set up user credentials etc) you have to write tests for those parts in that particular language. But the bulk of the API you can test in whichever language you prefer, then just assume the code generator is doing its job and creating a compatible API in the other languages.