I went ahead and built such a unit. Actually, I've been working on it for the past 6 month or more.
GPS receiver feeds 1 pps into PRS-10. User output comes from PRS-10 through buffer/distribution amplifiers. Today, I am doing preliminary tests using TICC + LH setup. Of course, effects of gps control won't be apparent for months but as of now, FTS4040/A (Cs) and my unit are neck to neck. Adev at 100 sec are 1.x E-11. At 500 sec, both 5 or 6 10E-12. I'm rather impressed....
I believe the default loop constant on GPS steering is something like 24 hours. So it's a very gentle pull. It has been turned on for about 2 weeks. My Cs is, unfortunately, quite old. Other than it locks effortlessly, I don't know it's accuracy or stability. I'm going to have to trust it's at least stable, since it can maintain lock for as long as I keep it on.
That's probably the way to go - just get familiar with logging to TimeLab and LH, and using the results.
Since the TICC has 2 channels, you could use the 4040 as the reference and measure it against a rubidium and GPS to get some idea of its shorter and longer term stability as well as its accuracy. This also effectively logs the rubidium against GPS.
To reduce the GPS noise you could use a GPSDO instead, but it might be best not to use a long time constant in the GPSDO as that can lead its oscillator affecting the frequency.
Even when you do not have a very good reference like a hydrogen maser, you can get a good idea of performance by measuring against several different references like quartz, rubidium and GPS.
If you can then also log the temperatures of the oscillators that you are using, as well as the air pressure and humidity, you will then be able to work out their environmental sensitivity.
In the case of a Rubidium oscillator the ambient conditions often dominate the performance. Ageing may be about 1E-13/day, but air pressure sensitivity could be up to 1E-13/mbar, and the TC could be over 10E-13/°C.