Looks like there were a couple of different versions of filament (T1) and Plate (T2) transformers in the LK-800A. In the doc that BJ Radionut linked for the LK-800, T1 & T2 look like this:
Where an output winding on T2 is rectified through D1 and providing 12vDC @ "A". On the QSK board, there are 2 Orange wires bringing in the 12vDC connecting to all of the high side of the relay coils (pin 4) in parallel, including the vacuum relay coil. The low side of all of the coils "B" are connected in parallel to the 2 Purple wires:
For this version there are 5 relays on the QSK board.
If this is what you have, 3 wires output on T1 and 4 wires on the output of T2, then you need a 12v Coil in the RJ1. If that is true, then someone replaced RJ1 with the wrong coil. It would certainly explain why it is flaky - using 12v on a coil designed for 26.5v.
On another version of the LK-800A (later?), it shows this for T1 & T2:
Where T1 has the additional output winding that generates 12v and +20v AC. The center tap is rectified through D1 on the power supply board and provides 12vDC @ "A". The high end of the winding (red wire) goes to connection 11 on the QSK board, where it is rectified by CR2 (1N5408). C2 (1000uF 50v) should bring the DC voltage up to 22~24v going to the high side (pin 3) of the coil. Pin 4 is the low side of the coil and it goes to Q1 (2N2222A) which switches ground:
This version has 3 relays on the QSK board. For this version, RJ1 should use the 26.5v coil.
TLDR: I bet $5 that someone (Jennings?) mismarked the relay (stamped 12 on it) and the factory or someone else installed it thinking it had the 12v coil, when in reality it has the 26.5v coil. It sounds like the LK-800 that you are looking at is the first version that matches the schematic that BJ Radionut linked.
Sorry for the confusion, I didn't realize there were 2 versions of the LK-800A.
I learned something new today...