I've owned a 4001 type from '75 I think it was. If your a fingerstyle player like me, I would not recommend them as the pickups are so high output, and crackle to the touch.I should also mention, Birch's builds vary massively - some have neck so thin they bow from slight string tension and other's are like a baseball bat - try before you buy!!
There's an SG bass in Electric Ladyland, Bristol - it's in black finish but the owner wants close to 2 grand for it and you have to go see him in person - no online shop. He has a double neck guitar/bass 4001 too but it has a repaired neck & headstock. I personally wouldn't pay more than a grand for one, but it's cool that they are vintage, custom shop basses and you can get that Geezer Butler tone straight from the plug in. The truss rod requires a 16 Box Spanner - I had to keep taking mine back to Overwater because it had so little play in over the years.
If you find a decent one for sub £1k i'd keep it, my guitar tutor taught me on an SG doubleneck guitar and a green 4001 bass; in my first band, the guitarist had a J1 too, so they're a bit of a nostalgia trip for me. Hope this helps!! Feel free to ask more questions! There's a couple of nice guys on here that own Birch's, hopefully they'll see this thread