Your unit looks basically healthy. Even when it loses lock, the Clock Signal (Cesium beam level) is basically constant and the level is good. Even the Zeeman Signal stays good. The only thing I'm questioning is the OCXO CV. It should be between about 0.5V and 5.0V. It's been a while since I looked at mine. Is there a coarse tuning adjustment on your oscillator? If so, try tweaking that to see if you can get that voltage to rise. OCXOs drift, so it wouldn't surprise me if it needed a tweak to bring it back where it should be. If there isn't a coarse tuning adjustment, I'd recommend removing the oscillator from the unit and powering it on the bench. Leave it powered for a week, or a month, and see if it drifts back. Don't try this while the oscillator is in the 4040A. There's no way to have the oscillator powered without using up your precious supply of Cesium! Actually, it would be useful to power the OCXO on the bench and run the EFC from one end to the other to see what the frequency range is. However, if the OCXO needs a lot of time to drift back where you want it, you'll probably have to do something so that it's at a reasonable frequency after a short warmup so that you don't waste Cesium.
Two other things you should check are the power supply capacitors under the metal lid and the outputs of the STEL-1173. You might consider adding a small heat sink to the 1173. I don't know if it will help, but that chip is prone to failure and does run hot. It can't hurt. You should also check the outputs of the 1173 to see if it's started to fail. You mentioned that yours seems to be a more recent unit so that might not be a problem for you - yet!
Regarding the tube testing, yes it's possible. I discussed it in the other thread and posted the results for my tube. You need a synthesized generator in the 12.8 MHz area that's referenced to your Rb standard, a DMM that has an input impedance of 100M or more, and a computer that can control the generator and read the value from the DMM. If you're a real glutton for punishment you could take the readings manually. Disconnect the tube's output and connect the DMM to the tube. Configure things so that the tube sees a 100M ohm load. Connect the generator in place of the 12.8 MHz source. Sweep the generator from maybe 12.5 to 13.0 MHz and measure the result. You should see something like the graph below. I stepped the generator in 10 Hz steps. Now, I wish I'd used 1 Hz steps.
This isn't a perfect test because while you're doing this, the processor is going crazy trying to change a bunch of parameters to try and find the signal that you disconnected. But, my results seemed reasonable. The level was significantly lower than you'd get from an HP 5061A Cesium, but the 5061A manual includes a calculation (paragraph 5-164) to check the beam signal quality. According to that, my tube is great - if that calculation even makes sense on a different unit.
Ed