I realise that I'll need a scandoubler, but that will still mean a monitor that can sync to a 31kHz horizontal signal. Is there anyone actually using a currently-available LCD monitor in such a way? Would you recommend it? I'm keen to use a flat panel screen so I can keep the space requirement down, and to use a monitor because tvs are licensed in the uk.
This may not be what you're looking for, what I've done is plug the RF video from the Amiga into a Dell LCD monitor that has a standard composite video RCA input. ( It's an older 24" LCD monitor from Dell that isn't sold anymore, but this is it: http://goo.gl/DzWd. Of course there are now many multi-input monitors on the market, but not so many with composite video inputs.)
A really nice feature of this monitor is the picture-in-picture mode: the video from the Amiga can be placed in any corner of the screen. This works out well, since that small size matches (roughly) the resolution from the Amiga anyway. You can have the video take up the entire display, of course, if you love big blocky pixels. :)
Anyway, for me, this has been a good one monitor solution for using my PC and Amiga at the same time.
I think you'll need something like this. It's not cheap though, but it should help in finding something that does the same thing, but cheaper.