http-api: T3616: update for strict content-type check in FastAPI 0.65.2

FastAPI 0.65.2 checks content-type request header before assuming JSON,
closing a well-known loophole. This requires a modification of the code
providing backwards compatibility of multipart forms.

(cherry picked from commit 3a9041e2d4d4a48ba7c01439e69c5f86a4a850c2)