Lock in amplifier reference signal
I wouldn't really trust it.

As you have pointed out, the whole point of lock-in amplification is that you know that the frequency and phase of the signal used to modulate the physical system. Two oscillators which are not phase-locked to some sort of master reference will drift apart from each other. If you plot the measured signal over time you will see it vary sinusoidally with a frequency equal to the difference in frequency between the two oscillators. It doesn't matter how close you get them to the same frequency, they are certain to drift eventually. This is why as you increase your time constant (ie: tighten the passband of the lowpass filter) the signal starts to vanish.

So, about all you can do is basically just use the lock-in amp as a bandpass filter/envelope detector. You'll need a dual-phase lock-in amp for this. If you have a dual-phase lock-in amplifier, you will have an in-phase and quadrature component output. Mathematically, you can consider their output to be a complex number encoding the magnitude and phase of the measured signal.

Since your oscillators aren't locked, you pretty much have to throw out the phase and just look at the magnitude. That'll just be the vector magnitude of the in-phase and quadrature signals:

$$ \left | s(t) \right | = \sqrt{I(t)^{2}+Q(t)^{2}} $$

Some fancier lock-in amps have a button for this, which gives the vector magnitude as an output.

This may or may not work for your application. You will lose a lot of the noise immunity advantages you get from synchronous lock-in amplification.

Personally, I would have your engineer crack open the device under test and probe until he finds the clock signal, then solder on a wire, but that's just me :).