I can’t say for certain, but from my days of looking for green lanes there not a lot between BG and Redditch off-road wise (to use as direct routes).
Also I’d consider going straight into the lanes from Bromsgrove towards webheath, rather than toward Tardebigge, unless you looking to purposely add miles to your commute.
Once your into Redditch it’s a different story, it’s got lots of cycle friendly paths, and has a surprising amount of small woods and copses to explore, nothing too rad so your CX would be fine.
Not that it’ll be obvious, as the place is a maze, and can take a bit of time to figure out…
I regularly do a 15-18mile loop of the town, taking in most of the woods, as friends live over that way. I know of locals who will drive out of town rather than ride there, but it’s better than nothing & is on their doorstep.