Installing a ceiling fan, 1920s home

The cross bracing there is just to keep the joists from twisting and isn't required. You can cut it out.

Take extreme caution in doing so!

1. Vibrations from any kind of saw will jiggle lots of dust down on you and could crack the plaster on the ceiling.
2. MORE IMPORTANTLY: You have a wire right there and it would be very easy to accidentally cut into it.

Make sure that this wire is dead (breaker is turned off, test to be sure with a non-contact voltage checker) before working near it.

Odds are really good that in addition to the old-work ceiling fan brace you've got there (good job on picking that up!) you'll also need an extension of some sort in order for the ceiling fan bracket to actually reach the mount points on the box. Because of the height of the legs on your old-work support and the extra depth of plaster & lath, the box that comes with the support just won't be tall enough to be flush with the ceiling or slightly below. I'm not certain of the best, code approved, way of doing this in a retro-fit, so you may want to ask a new question about that.