[C++][FlightSQL][ODBC] Replace RapidJSON with JsonWriter #50911

Describe the enhancement requested

The Flight SQL ODBC module contains remaining RapidJSON usage for JSON serialization under cpp/src/arrow/flight/sql/odbc.

Replace the RapidJSON usage with Arrow's existing JsonWriter, removing the now-unnecessary RapidJSON dependency while preserving existing JSON conversion behavior and test coverage.

Additional context

Parent Issue: GH-35460.

Component(s)

C++