I apologize in advance for the lack of meaningful formulas or calculations, I am not a mathematician and am using excel to try to compute everything. I think there is likely a simpler way to approach this but the only info I have found anywhere close to similar to my problem leads to Euler angle matrices.
I have a 3 axis geared tripod head that I am using for astrophotography, it is mounted on a motorized sky tracker (its axis of rotation can be ignored since it only is used to stay centered on deep sky objects) and a two axis wedge. The X axis on the wedge is just for fine-tuning yaw when aligning with the celestial pole. Since the sky tracker has to be perpendicular with the celestial pole to track with the sky's movement, the pitch axis of the wedge needs to be set to the GPS latitude. Let's simplify and say it is set to 35 degrees (although I have been using 325 degrees in my Euler angle calculations since those are derived from counter-clockwise rotations). Now my issue is my 3 axis tripod head is now at this pitch angle offset and I need to be able to make precise movements in relation to Right Ascension and Declination coordinates in the sky. Software with the camera will tell me, based on the stars in the frame, what my (RA, Dec) coordinates are and based on the target coordinates I am trying to photograph I can compute east/west and north/south angular offsets in degrees.
Let's say, if the 3 axis head was on a level surface and I needed to adjust 2 degrees north and 5 degrees east I could adjust the pitch and yaw axes respectively. However with each axis being offset by the wedge's pitch angle, I want to calculate the individual angles of rotation needed for yaw, pitch, and roll axes of the 3 axis head in order to make the 2 dimensional adjustment in regards to the sky.
I have tried solving Euler angle matrices with respect to the pitch offset, however I have not been able to figure it out. Any help, especially if it would lead to an easier and more straightforward way of calculating it, would be greatly appreciated.
Once you're polar aligned properly, your tripod head's yaw will correspond to Right Ascension, and pitch corresponds to Declination. So you can adjust those angles directly with these two axes. There is no angle rotation calculation to perform, since you adjust those angles directly with your head axes.
The purpose of the wedge yaw and pitch (we usually call those azimuth and altitude in astronomy lingo) is to polar align your mount. You seem to think that the wedge altitude adjustment messes with the RA and Dec angles. But in fact, this adjustment makes the head line up with the celestial axes, since you're aligning with the pole hence the celestial frame. Once done, the head base becomes parallel to the celestial equator, so its yaw will move your camera in Right Ascension, and its pitch will move it in Declination, and the last axis (roll), will rotate your picture frame.
Here is your setup with added graphics to help you understand:
Knobs identification: