I'm not promising this will help your situation, but I for one refuse to (try to) solder to the back of a pot under any circumstances. I will find another way. You can try to find ring connectors big enough, but I usually just solder the wire to the washer that was going to go on the shaft already.
But also, the connection to the back of the pots is really just a shield connection. While it might be convenient and it definitely is common, it's not actually best practice to rely on that shield connection to carry your circuit ground. From the point of view of getting the circuit itself to actually work, the important thing is to connect the pickup bottom wires, the bottom lugs of the V pots, and the bottom ends of the T caps to the jack sleeve. We want all of those connections to be solid and reliable, and you're much more likely to achieve that if you use wires or component leads and actual solder lugs.
How you do that is up to you, but maybe have the bottom wire of the pickup and its T cap go to its V pot bottom lug*, then jumper one V bottom lug to the other and stick the jack sleeve wire right there.
Now your circuit works, now connect your shield wires (pot backs, bridge ground...) to the jack sleeve. Now it's as good as it's gonna get.
*by bottom lug I mean the "ground" lug, if you're doing the washer thing, this can be unsoldered from the case and hopefully bent back into place without breaking