While searching the forums for the answer to my question I discovered that the answer was an unsatisfying "just eyeball it". As my title suggests my question was how to reliably park a satellite in geostationary orbit above a specific spot beneath that orbit.
Well unsatisfied with the answers in that thread I came up with a method of "eyeballing it" that works extremely well and wanted to share. I don't know if there are others who have discovered this since the post I found was older but here it is regardless:
I'll warn in advance this is not a newb friendly explanation (though it probably could be), but it is very doable for those with moderate experience (i.e. most anyone looking for how to do this).
Edit: I decided to add a high level explanation before diving into this to help people follow the steps.
Strategy Explained - Essentially what this strategy is doing is finding the time difference between when you and your target are beneath the Ap and then calculating an adjustment burn to make at your Ap in order to have both of you arrive back at the position of your adjustment burn at the same time after 1 orbit, at which point all you will need to do is re-circularize.
Park in Geostationary orbit around the body in question without regard to what you are positioned above.
Go into map view and set the focus on the host body. **This is important**
Still in map view, align your view so that the Ap marker (closer to camera) covers the Pe marker (away from camera). You may need to adjust your zoom to fine tune this, it helps to start far out and align it then zoom in and re-align etc.... Because of the circularized orbit this will point to the spot roughly beneath the Ap (close enough for us). Hold this camera position until Step 5.
Steps 4a and 4b can be done in whatever order is convenient.
Wait (or advance time) until your satellite passes under the Ap marker and note the in-game time (referred to below as ST for Satellite Time).
Wait (or advance time) until your ground target passes underneath your Ap marker and again note the in game time (referred to below as TT for Target Time).
Next you need to determine the orbital period (referred to below as OP) for both you and your host body (they will be the same). You can use mods like MechJeb, Engineer, look up the value on the wiki (where it is referred to as the "sidereal rotation period" on the host body's page), or just click right behind your satellite on the orbit to get a good approximation. For fast spinning host bodies you will want a more precise answer as your margins will be smaller.
OK, now for some relatively simple math. Note at this point we have not performed any burns so you have as much time as you need to work this out, but you can also just pause the game.
ST-TT=Orbital Lead (OL)
OP-OL=Adjustment(ADJ) and I also advise you calculate ADJ/2=Half-Adjustment(HADJ)
Note that the time in game is on a 5h59m9.4s day cycle. For most calculations you can treat this as 6h, but if you are dealing with a fast rotating host body it may matter.
Set a maneuver node at your Ap and adjust it until the projected course has a time to Pe (seen by mousing over the projected Pe marker) equal to your HADJ. Depending on the sign of your OL this maneuver will need to be a Prograde (Negative) or Retrograde (Positive) burn. It helps to think of this like orbital rendezvous where when the sign is postive you need to catch up to the target and when it is negative it needs to catch up to you.
Perform the maneuver node at your next Ap and then re-circularize at the following Ap once you get back to that point (It will either still be your Ap or it could be your Pe if you had to raise your Ap).
Example: This is a real example from when I wast testing this out.
After achieving Geostationary orbit and aligning my Ap/Pe markers I recorded an ST of 23d5h27mXXs and TT of 24d2h40mXXs. The "XX" here just means "Don't care" because the orbital period (OP) on the wiki for this body (Minmus) is almost 2 days (1d5h13mXXs) so the seconds were not going to be an issue. Faster host bodies may be different though, for Minmus I used a day of 6h and subtracted a minute from my answer for each time I rolled over a day to help keep things close.
With the above values I calculated an OL=-3h12m which means I was ahead of the target and needed a prograde burn to let it catch up. An important note here is that adding and subtracting non-standard time units may be a little weird for some people so you will want to isolate things as best you can. In this instance I saw the ST was 32m from rolling over to 24d, so I added the 32m to the 2h40m past 24d from the TT to get -3h12m. Most of the time I ignore the sign and just work out the difference in time between the two and determine my burn direction by if I am ahead or behind by visually looking at things. Once you understand the principles of this method you probably will also.
With an OP of 1d5h13mXXs I subtract -3h12m (e.g. we add 3h12m) and get an ADJ of 2d2h22m (funny how that came out) giving me a HADJ of 1d1h11m. From here I just fine tuned a prograde burn to occur at my Ap until the projected time to Pe on that maneuver was 1d1h11m. Then all I had to do was execute the prograde maneuver and then re-circularize on the next pass at the Ap (edit: former Ap in my case - after my adjustment burn it became my Pe)
Using this method I was able to perform 2 minor maneuvers of 9.4 dV total to get right on top of my target and get credit for my contract.
After the fact I thought about the formulas above and it might be more canonical to write it as TT-ST=OL and OP+OL=ADJ but they both get the job done (and adjusting it would have required changing my references to the implications of postive/negative OL values in the text increasing the already high chance I mess something up).
Anyways, there are probably more steps than is strictly necessary, the actual technique is not too complicated if you can grasp the principle underlying it. I hope someone finds this useful.
Edit: Made a large edit to add some formatting as I often find giving the eye something to latch onto helps when referencing different parts of the text. Also a flair of color to clarify an easily confused bit doesn't hurt.
