Folks,
Thank you so much for the replies and recommendations. Already there with the flatwounds, foam and vintage basses (a Fender Jazz and a Baldwin Burns Vibraslim). Although that's pretty good for a live sound, I'm really trying to refine the recorded sound. I tend to DI and split it as mentioned, and the use of valve compression is a given. There's also a lot of valve and tape emulation available for computer recording, modelling Pultecs and desks etc., which I really like. Some sounds are hard to take apart though. Like John's Children recordings. Great bass on there.
Any other advice, anecdotes and wisdom is gratefully received.