I discovered the answer in this video on Adobe.tv:

Working with DSLR Video in Lightroom 4

It's pretty nasty:

1. Play the video in the Library module and pause when you get to a frame that you want to use as a basis for editing.
2. On the video control strip (the black strip with the play/pause button and the timeline scrubber), click the little rectangle icon (just to the right of the playback time) and choose Capture Frame. This generates a JPEG of that frame and adds it to your catalog alongside the video.
3. Select the JPEG, switch to the Develop module and process as normal.
4. When you're done, sync the JPEG's develop settings back to the video using any of the usual methods (select both in Library and hit the Sync Settings button, or copy/paste).

Lightroom will copy the settings that can be applied to video (e.g. cross processing, saturation, etc.) but ignore the ones that can't (such as crop, spot removal, post-crop vignetting and grain).

It seems the only additional features available via this route (on top of what's already supported in Quick Develop) are:

• Tone Curve
• Saturation
• Color Adjustments
• Split Toning
• Process Version
• Calibration

Added to the fact that there's no History panel for videos (so you can't retrospectively roll back individual changes), I think video editing in LR4 should very much be considered a novelty bolt-on rather than a fully integrated feature.