No reason why not. Makes perfect sense, as per the Allan Woody bass.
All of the commercially available basses (past & present) with a pickup in that location will generate a rounder sound than pickups further from the neck.
Using a Jazz pickup, you might have to be more careful in selecting one that supposedly has a "vintage" tone, with corresponding care in the choice of tone capacitor, to get the exact sound you're after.
Also, be aware of possible problems due to the narrow spacing of the strings at that point.
When Roger Glover of Deep Purple put a pair of Jazz pickups next to each other and close to the neck on a Rick, they were very slanted so the pole pieces would line up with the strings.
You might be able to find info on which specific songs were recorded with that bass?
I would use a humbucker, and a relatively narrow one at that.
If it has to be a Jazz pickup, are there any with a blade pole? Never an alignment problem with those.