My first suggestion would be to see if you can find a way to SOLIDLY mount the indicator directly to the headstock. If you follow the path from the base, on the carriage, through the ways, you've got room enough that you probably can influence the indicator just by pushing on the back of the spindle.
The way you have it, when you push on the spindle, ANY movement within the whole of the machine is going to show axial play, whether it's real or not. You should be pushing/pulling pretty firmly on this. The way to test it is this- Run the adjusters tight by hand. As tight as you can get 'em. Rotate the spindle by hand, then, if needed, use whatever tool to tighten a little more, until you can feel some interference when you try to rotate the spindle. Without a chuck mounted, the "rules of thumb" don't work, but that's fine. It SHOULD rotate by hand, just grabbing the end and turning it, but it shouldn't be a fight. Just tangible (VERY tangible) resistance, but one hand with reasonable effort should do it. (Don't power it up like this, but slowly, you're not even remotely coming close to any dynamic loads... No worries...). At that point, you have enough preload that you, yourself, are not capable of moving the indicator IF (and it's a big if), IF the indicator is mounted to a solid point for measuring. So now, with everything "too tight to a degree that you can be 450 percent sure that there's ZERO axial movement under hand force- See what your reading is when you push and pull. That'll prove or disprove the mounting situation that I don't approve of, but "maybe"... It's fine? Science wins over speculation.
Now, with the indicator in a PROVEN location and setup... Back the adjusters off until you can measure axial play, and start the process of incrementally tightening the collar.
As you snug the collar, the reading should reduce. Repeat that in very, very small increments, approaching zero. As you watch your reading, and move in small, EQUAL increments, you'll probably be able to feel out how much turning gets you one one thousandth of an inch less axial play. When you get it down to one, you'll have one more equal "move" to be at zero. And one more would get you one thousandth of preload. And two more would get you two thousandths of preload.
For some ballpark to get you started, if the adjuster is on a 16 tpi thread, then one thread is one rotation. So one full rotation, 360 degrees, is going to get you 0.062 thousandths of preload. (Don't do that...). Call that 0.060 thousandths. (Plenty close enough, you're gonna divide this error down to nothing). Divide by four for 90 degrees, that's 0.016 per quarter turn, 0.008 per eighth turn... You can see that's going to get fiddly.
There's a spline or a gear or something on the back of that spindle that's got "some" number of teeth. I'd suggest taking that 0.060 per revolution, and dividing that into whatever number of splines or teeth you find back there to use as a reference. So if you find (for an arbitrary example, I have no idea), if you find 20 "teeth" back there, 0.060 per turn divided by 20 teeth, that's 3 thousandths per tooth, so one third of a tooth space would be your increment. Much more managable.
So yeah... It's not a quick measurement, but once you validate a steady indicator, it's just a matter of iterating to get the feel. Then it speeds right along. And when you're done, run the lathe for a time. VERIFY that the bearings are at or just below the point where you can
If you find some number that doesn't work at all, like seventeen teeth (you won't find that...), but it doesn't divide well, it's fine. You're moving in small increments, and resetting your counting point every thousandth (or so) from the measurement on the indicator. So any error is "one time", they won't stack up, and will be well within any reasonable tolerance.
In the end, if they want 0.001 to 0.002 of preload, I'd say that you'd be "fine" at a half to three, and if you were off a little further than that (within reason)- Your grand kids might not live long enough to know the difference. I'd shoot for the lower side honestly, and do my best to hit 0.001. These bearings will not be challenged all that much (relatively speaking), so there is no need to maximize the preload, just so long as it's there.
And, as you're new at this (and maybe even if you weren't new at this), I'd validate it. Run the lathe again for a half hour. The ideal place to be on something such as this is at a middle to two thirds speed, you want to be at or just below the point where you can feel a temperature difference in the headstock casting with your fingers. Just barely notable if you're looking for it, but might miss it if you were not looking for it is perfect. That point or less, and you're done.
