feat(plex): Live TV channel + EPG sync after generation

[aresthegodofwar]

Addresses #696.

Summary

After each generation run, refreshes any configured Plex servers so newly created and removed channels appear in Plex's Live TV guide without a manual rescan — the same refresh-after-generation pattern already used for Emby, Jellyfin, and Channels DVR.

• Settings → Media Servers → Plex: URL, token, a discovered DVR/device picker (GET /livetv/dvrs), and an optional Dispatcharr channel profile to scope which managed channels get pushed to a profile-bound device.
• PUT /media/grabbers/devices/<key>/channelmap is the only call needed. It's a full-state-replace on both the enabled-channel set and the per-channel EPG-binding params, and issuing it is what makes Plex actually refresh its guide.
• Every enabled channel outside Teamarr's own managed range is resubmitted with its own current EPG binding on every run, unchanged, so channels from other tools sharing the same device are never disturbed.
• Keyed on each entry's deviceIdentifier (Dispatcharr's physical channel number), never Plex's own channelKey — see findings below.

Findings from live testing (relevant to the discussion on #696)

Plex exposes no documentation for the DVR/channelmap endpoints used here, so most of the above came from testing against a real Plex + Dispatcharr HDHomeRun setup rather than from docs:

• channelKey is not a stable channel identity. It tracks whichever EPG entry is currently matched in Plex's Channel Matching UI and can diverge from the physical channel number after a manual remap. deviceIdentifier is the field that stays fixed to the actual Dispatcharr/HDHomeRun channel number — everything here keys on that instead.
• The channelmap PUT is full-state on both params, not just channelsEnabled. A channel listed as enabled but missing its own channelMappingByKey/channelMapping entry gets disabled anyway — there's no safe delta here, every enabled channel's binding has to be resubmitted every run, foreign ones included.
• POST /livetv/dvrs/<id>/reloadGuide does not reliably refresh programme content. A channel reassigned to a different event (same number, new programme) kept showing stale guide data through several reloadGuide calls. The channelmap PUT is what actually triggers a full guide refresh on Plex's side — it's the same request Plex's own UI sends when you open a tuner's Channel Matching screen and hit Save with no changes. reloadGuide isn't used by this implementation at all.

@egyptiangio @pumacustom0c — flagging in case any of this overlaps with what you've each found; happy to compare notes if there's a better way to combine efforts here.

Test plan

• ruff check teamarr/ tests/, pytest tests/ -v (3493 passed), cd frontend && npm run build all green on this branch rebased onto current dev.
• New unit tests in tests/integrations/test_plex_client.py covering the fetch-merge-write channelmap logic, the deviceIdentifier/channelKey divergence, and URL/param handling.
• Manually verified against a live Plex Pass server with Live TV/DVR pointed at Dispatcharr's HDHomeRun emulation: new Teamarr channels appear and get EPG-bound, channels removed from Dispatcharr drop out of the Plex lineup, and channels configured outside Teamarr (manually added, sharing the same tuner) keep their own guide data intact across repeated runs.