Belt-and-suspenders complement to the cached-system-prompt inheritance:
pin session_start and session_id to the parent's so any code path that
re-renders parts of the system prompt (compression, plugin hooks)
still produces byte-identical output. The cached-prompt assignment
already short-circuits the normal rebuild path, but these pins
guarantee parity even if a future code path bypasses the cache.
Idea from simpolism's reference PR #25427 for #25322.
Co-Authored-By: simpolism <32201324+simpolism@users.noreply.github.com>