OK, so it's a b+W head, not a dichroic so it doesn't use the 24v lamp. The idea of using a diode to drop 120v to 84v seems like a bit of a kluge to me but I know it was common practice in the US. The ideal solution would be to find a 220v lamp which would fit the enlarger's socket, chop out the diode and run the whole lot at 220v, but my knowledge of enlarger lamp types is too limited to be able to say if that's a viable option.
I don't think you would need as big a 220-110v transformer as 1KW, the Analyser's consumption is negligible so you're really only running the 250W enlarger lamp so 500W should be more than enough.
There is no problem with the frequency difference, our timers have their own internal clock and do not rely on the mains frequency.
Edit: looking on the web I found this which suggests there is a stabilised power supply for the 82v lamp - if you have that, ignore what I said about the diode .