Implement transparent client pattern and middleware error handling

[andrii-balitskyi]

          Not in scope for this PR, as it would make it too large and this can focus on introducing the client into the existing module, but we should follow a transparent pattern for using the client [as was done for Python](https://github.com/seamapi/python/blob/cb26ee4f9544fafae3e3614f1b02eda41a3445a6/seam/routes/access_codes_simulate.py#L23-L27) and move error handling into middleware.

Originally posted by @razor-x in #134 (comment)