Building your own is a good way forward. I had a 90s MIJ Fender Precision body fitted with a Mighty Mite Jazz neck and, eventually, a Fender original pickup. That was delicious and I regret not keeping it as it did all the things my subsequent basses have done.
Fender Jazz Bass Special models have Jazz shaped necks but Precision bodies. P/J pups. Usually sell for around £400.
Precision Lyte has an uber slim neck, very shallow. Sold mine now but there are a couple for sale within your budget.
Squier Silver Series precision basses were the last of the MIJ Squiers from the early 90s, generally exceptionally good, and had a 40mm wide neck.