[Drizzle Kit]: Extend api (#4999)

* feat: Add external kit api to start Drizzle Studio

* feat: Separate API build process and fix ESM dynamic require handling

* fix: Update import paths for Relations in api.ts, pgImports.ts, and studio.ts

* dprint

* fix: Remove '-pooler' suffix from NEON_CONNECTION_STRING in Vercel client tests

* dprint

* chore: update version to 0.31.6 and add changelog for bug fixes