An update - I spent ages looking for a 2nd original pickup, but in the end a pair of EMG 40Hz 5-string pickups came up and now they're fitted to the Spirit 5; I did have to do a bit of surgery with my Dremel and now need a couple of surrounds to cover the gap either side of the pickup bay, where the EMGs are just slightly narrower.
The bass sounds really good now - surprisingly good, I'd say!