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You have been spoiled! Most preamps have no low battery warning. Some just go dead and can leave you high and dry mid-song. With others the sound quality deteriorates shortly before it dies. Either way always have a spare battery to hand! I put a bit of masking tape on the battery cover and mark the date when the battery was last changed. That at least gives me some idea of when the next change is going to be due. If you have (or can add) an active/passive switching option so much the better.

This sounds about right to me, amps can pick up all sorts of interference from nearby mobile phones

I use MuseScore which is free and does everything I need it to. MuseScore has a feature called 'voices' which I think allows for simple vocal scores - https://musescore.org/en/handbook/voices

Re 70s J bridge pickup position - my '72 measures 377mm from 12th fret to centreline of the pickup poles.

The 'regular' series wiring for a Jazz (with the two pickups configured as a very wide aperture series humbucker and single volume control) gives a thick and chunky sound which is very usable but a bit one dimensional. It's quite a bit louder and somewhat darker in tone with less sparkly highs than a standard Jazz sound. The phase cancellation in series mode is actually much less apparent than with both pickups on full using standard Jazz wiring (which is effectively a wide aperture parallel humbucker). I have a medium scale Jazz with a series switch. It's a useful alternative tone to have on tap but in practice I don't use it that often. A Jazz just sounds better when it sounds like a Jazz to my ears!

Useful tip, but won't a hairdryer stir up a lot of dust and swap one problem for another?

This (old) TB thread mentions that British Audio Services (Nashville) sell a CD which includes the AH600SMX schematic for $15. May be worth contacting them if you can't find it elsewhere. https://www.talkbass.com/threads/need-schematic-for-trace-elliot-ah600-smx.389111/ Or try a UK Trace repair centre such as https://www.proaudioservice.co.uk/bass-amplifier-repair/

If the bass is a P with an added J then generally the P is in the exact same place. If the bass is a J with a P neck pickup then it's generally a little bit north of where it should be. The other common positional variation with P/Js is having the J pickup in the 60s or 70s position. I agree with Misdee about Yamaha BBs being among the best sounding P/Js. Particularly models with a hot blade J bridge pickup like the BB1024 etc. A regular J pickup just sounds weak when matched with a P and isn't man enough for the job IME.

It depends how the P and J pups are wired. If it has a single Vol and Tone and a 3-way pickup selector switch (P, P+J. J) then, when in the P position it will be exactly the same as a P bass. On the other hand, if the pickups are wired Vol/Vol/Tone (or Vol/Blend/Tone) then the resistance of the additional pot changes the pickup loading and the solo'd P will sound slightly darker. It's not hugely noticeable, but there is a subtle tonal difference. If you want to retain the original tonal range of the P pickup then wire it with a selector switch.

Fender vintage saddle screws are generally UNC 6-32 x 1/2" (or length to suit)

I'm glad I wasn't there lol!

If no one here can help then there's a long and active thread on TB about Ampeg scroll basses - https://www.talkbass.com/threads/love-of-ampeg-and-other-scroll-basses-part-2.1044767/page-68. Lots of AEB-1 owners on there who may be able to help. Visual comparison with P-bass below might be useful as a guide.

Have you tried swapping the hot/cold wires on the Jazz pup? That's an easy way of checking for a phasing issue. Or try reversing the phase of one pup channel in Logic as in vid below

Ah, how annoying, wonder why they did that!

Excellent!

The Mustang, Bronco and Jaguar SS are all 30" scale so it is possible the JSS neck will fit. The pocket and mounting hole positions look very similar.

The headstock reshape was a brave attempt lol! The SD single coil with the bigger pole pieces is probably quite similar to my Sentell pup. There's something about the sound of a bass with one single-coil that is so open and natural it just feels 'right' to me.

I've spent a lot over the years in my quest for 'the special one' and have owned exotic and beautiful instruments from Zon, Rick Turner and Rob Allen as well as many other expensive-ish basses. More recently I've been playing around with budget basses and have been very pleasantly surprised! My latest one is this Squier Bronco which was professionally modded by the previous owner with a Hipshot bridge, Hipshot tuners, new pots/harness and a Sentell 'Big Sis' handwound pickup (http://www.sentellpickups.net/bass.html). An ebay special for the princely sum of £150. A bit on the steep side for a used Bronco maybe, but it's a little gem! It feels and plays well above it's price point and the Sentell pickup is gloriously rich and sensitive to playing touch. I love it! Anyone else out there with something rather 'special' at the budget end of the GAS richter scale? Post pics if you can!

Ah, sorry I didn't realise the yellow wire joined the two diagonal lugs. Looks like that's just a jumpwire which hardwires the two coils in series then. Which means the two connecting points you need are as in pic below. Give that a try that and see if it works.