I doubt you have the tools to rebuild one. Pullers to get the pinion bearing inner race off, presses to fit the new one, dial gauges to set the backlash, shimmed circlips for the side bearings.....I could be wrong though. I've rebuilt enough diffs to know it's not worth the bother on such an inexpensive differential.
A 2.5 uses a bigger Type 188 diff, a different thing and it's very rare for one to fail.
The ratio is 3.38 (manual box) and the part number for a Z4 is 7551912. Or 7551857 for a 3.73 auto diff.
It's another E46 diff with a finned rear cover.
The easiest to find is a 3.38 from a 320i with the 188 diff.
Or a 3.23 from a 325i Auto.
The older M54 2.5 diff is a 3.23 ratio and close enough that you won't notice. Like this one:
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I bet £150 will buy it.