Snapshot Content-Type and body while the client context is still
active so pooled connections fully release on exit. Previously the
read happened after `async with httpx.AsyncClient(...)` returned —
which works today only because httpx eagerly buffers non-streaming
responses; a future refactor to `.stream()` would silently read-
after-close.
Part of the #18451 connection-hygiene audit. Salvage of #18502.
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