OK, Attilio, here you go:
I looked at the LO drive circuits and circuit descriptions and calibration info in the TR503 and TR502 manuals. I think you've been hung up on this 2LO BPF adjustment issue, where you're manually driving the LO with an external supply. Forget about that for now. You don't need to redesign the circuits to get higher tune voltage - it's already capable of reaching around 17 V, maybe a little more, which hopefully is plenty enough. At the plus extreme, U365A's output should rail at somewhere around +18 V or so (you should look up the NE5558 specs to see just how high it can get), since it's powered from +20 V. Q420 and the current limiter should drop about a volt, so the output will be able to get to around +17 V, and intrinsically limited there. Q445 and Q430 form a search oscillator to make sure the tune voltage sweeps around until phase lock is achieved, but apparently only kicks in if the tune voltage drops below +3 V. In the BPF tuning procedure with external supply drive, it says something like "do not exceed 20 V," so the capability of the circuit is within the limit.
In your test setup, you already have the PLL open-loop since you've run the IF into the counter, and the PLL tune voltage is disconnected. So, using your current setup, with the LO powered from the +15 V external supply, and measuring about 2095 MHz on the counter, check the PLL output tune signal with a scope - it should be either stuck high near 17 V, or sweeping up and down between there and some low value around a couple volts.
If it's stuck near 17 V, then that's perfect, just what you want for the next step. Now shut everything down and disconnect the external supply, and reconnect P446 so the PLL will run the LO. Fire it up and observe the frequency on the counter - it should be stable and well above 2095 MHz. Continue to monitor the tune voltage on the scope, and make sure it's still up around 17 V, not sweeping. If it is sweeping, then disable the search oscillator by grounding Q445's collector. You should now see the maximum tuning frequency available, on the counter. Hopefully, it will have a comfortable margin, maybe 2100 MHz or higher. If so, then shut down again, and hook everything back up to normal, and connect the 2LO from the 7L13. Continue to monitor the tune voltage. Fire it all up, and see if the PLL locks, indicated by the tune signal being constant, somewhere near +15 V. If it is, then you've lucked out, and the 2LO BPF is close enough to get a good signal through, the IF and PLL are working, and the LO is running around 2095 MHz. The rest is then a matter of perfecting the changes and tweaking it up. At this point it would be good to shut down, and solder an 18 V, 1 W, 5% Zener diode onto the LO's power supply/tune connection, from there to ground, to serve as an absolute clamp around +18 V. It will also provide reverse voltage protection. The physical location can be at the module (best) or wherever convenient, as long as it's on that line, and can't be disconnected accidentally. This will give some protection against accidents during any situations where external driving is used - later during tweaking - or if a failure in the tune driver (like if Q440 were to short) lets it go over-voltage.
Whether this all works, or doesn't, please report the results, and we'll see what to do next. Good luck.