These issues are unrelated.
You can introduce a phase step and watch the loop response, but you need a proper loop filter to separate the phase response so you can measure it. Very few loops have the required filter. It is usually easier to introduce a small frequency step and look for the proper loop damping response. The damping waveforms are illustrated in innumerable engineering and mechanical sites.
Manually optimizing the filter will have no effect on the deadband. This is caused when the loop amplifier is too slow to pass the narrow pulses from the phase detector when the error is near zero. You can introduce extra delay in the reset path to make the pulses wider, or use a different loop amplifier to eliminate the deadband.
Usually the best approach is to use the design that gives the minimum reset delay since this minimizes the phase error while the up and down current sources are both on. You also need a good LC filter at the input to the error amplifier to avoid overloading the input stage and introducing extra phase noise.
You are going to have problems modeling this in SPICE. The ratio between the loop filter and the carrier frequency is too large.
Rubidiums are usually very noisy and would be unsuitable for multiplication to 10GHz. The SRS PRS10 is one of the few that have a low phase noise oscillator suitable for frequency multiplication. They are not too expensive:
If that is too much, you would get far better results using a quality OCXO with a very loose lock to 1PPS from GPS. Loose lock means a very low bandwidth PLL. You can get a Morion MV89 OCXO on eBay real cheap. Be advised the output coupling capacitor often goes open and it will no longer drive 50 ohms. I have several like that. There is a way to open the case by unsoldering the case and inserting a thin titanium wedge between the case and the bottom as you go around the case. Then you can change the cap and resolder. There are few web sites that show the MV89 disassembled and illustrate the oven construction. Unfortunately my Windows just crashed so I cannot search for the urls.
However, if you don't feel like fixing the cap, there is usually plenty of signal left to drive a high impedance load, such as a quality zero cross detector mounted directly on the unit.
Also know the MV89 is basically a 5MHz OCXO with an internal doubler to
10MHz. If you multiply to a higher frequency you have to add extra filtering to remove the 5MHz sidebands. This could be done with a low bqandwidth PLL.You can check the Time-Nuts archives for detailed information on Rubidium, OCXO, GPSDO, ADEV, DMTD, and all other topics related to accurate low phase noise frequency stabilization.
You can search the Time-Nuts archive using this url plus your search string: