Just a guess, but...
I've repaired several of the 1345A displays. The most common failure are the 2.2 uF tantalum capacitors (usually yellow cylindrical) on the A1 xyz amp/stroke board. One or more shorts out and takes out one or more of the low voltage power rails (+/-15V). Clearly not the case with your unit.
But once, when I was trying to repair a 1345A, I started lifting one end of each tantalum until the short went away. I replaced the shorted cap and the 1345A sprang to life. However, I saw behavior like you are getting - distorted characters. Turns out that I forgot to solder all the leads of the good capacitors that I had pulled up. Once I did that, everything worked perfectly.
So perhaps you have an open tantalum capacitor.