ikay

Fabulous indeed, a great band!

If white is hot then red should go to ground along with the unshielded wire. The bare wire is probably being used to earth the polepieces or could just be a shield. The red and white wires will be from opposite ends of the coil so one will be hot and the other cold (ground). If you leave either of these unconnnected the pickup will effectively be open-circuit and you won't get a signal.

Or two core with a bare shield wire like this?

Main advantage is the low B makes an excellent thumb rest lol!

Have you tried giving the trussrod a small tweak (say 1/8 turn clockwise) to make sure it's under tension?

Front view shows neck alignment a bit better and looks fine to me.

Depends on how the cab is wired, whether it has a crossover and attenuator for the tweeter etc - https://www.talkbass.com/threads/anybody-remove-the-crossover-disconnect-the-tweeter.391127/

Unfortunately a Tele guitar pup is quite a bit smaller than a 51 P ...

The Music Locker (Ripley, Derbyshire) specialise in vintage brands including Burns and offer to help source hard to find instruments - http://themusiclocker.co.uk/product/vintage-rare-electric-acoustic-guitars-for-sale/?filter_make=burns

A nice little job for someone with a 3D printer perhaps...?

https://www.gumtree.com/p/guitar-instrument/vintage-1960-s-vox-clubman-bass-guitar-sunburst/1308913097

Has it always done this or is it something new, since you last changed strings for example? Could be a dead spot, but that's something you would probably have noticed before. To eliminate that possibility, try resting the headstock up against something solid like a door frame and then play the A# on the G string. If the dead note goes away (or shifts up a few frets) that would indicate a dead spot. If no change then it's something else.

I've had that before and just wicked a little thin superglue into the joint which fixed it.

Probably best to have a chat with Yamaha tech support and see if they can help - https://www.yamahamusiclondon.com/forms.php?form_id=2

Here's another take on it from Schorr Guitars with a floating pickup on side rails. Rather 'individual' on the aesthetic front! Ultra low profile pickups such as Lace that can freely move around between body and strings might be a bit easier to work with.

Apart from the Grabber and Westone Rail, other concepts I've come across in my google meanderings (I've been interested in this for some time) include a 'swinging arm' pickup, single slider (guitar) and even one pickup with slide-able pole pieces (Lieber EQ-4 - https://www.lieberguitars.com/history/)

This Warwick custom sliding dual SC pickup is very neat and amazingly flexible

+1 for a moveable pickup. I prefer the simplicity of single pickup basses but would like to be able to tweak the position to taste depending on musical circumstances. Would love to see your take on that. Something with a tonal range from HB series/parallel/SC split and positional range from Ric neck to 70s Jazz bridge would cover it!

This wiki about Nicky Wire mention a Fender Jazz with a custom third pickup (under Musical Equipment) - https://wikivisually.com/wiki/Nicky_Wire The old Clover Avenger had three Delano jazz style pickups, I've always fancied one of these (the new model has two soapbars).

Might be worth contacting SBS in Manchester - https://sbsav.co.uk/shop/spare-parts/behringer/