Is there a formula to calculate any star mass from Luminosity, Radius, and/or Temperature (K)? I know there is the Mass-Luminosity relationship, but I am wondering if there is a more accurate formula I can use based on the data points I have generated:

Absolute Magnitude (based on class and type of star in relation to H.R. diagram)
Temperature (same as above; in Kelvins)
Luminosity (calculated from Absolute Magnitude)
Radius (calculated from Temperature and Luminosity)

Basically the last 2 pieces I need to calculate are Gravity and Mass which is a pain because the one formula I have requires one of them to find the other. So far everything is calculated from another piece and so all of my values are in line with each other.
I could use the Mass-Luminosity relationship if I must, but are there any other formulas? I couldn't find anything that didn't require gravity. I doubt it, but there isn't by chance something to calculate gravity from my data points is there?
I think the issue you are going to have here is that the position of a star in the HR diagram - which amounts to saying what the (2 dimensional) spectral type is - does not just depend on its mass.
The Vogt-Russell theorem says that the position of a star on the HR diagram depends on its mass, its composition and crucially, how its chemical elements are distributed in the interior. Thus we know for example that we could find a 2 solar mass star on the main sequence with a spectral type of say A2V but it is also possible for a K0III giant to be a star of 2 solar masses.
So there really isn't any shortcut or quick formula that is going to give you a route to finding the mass based on a spectral type (or equivalently, based on a luminosity and effective temperature).
If you know the star is on the main sequence then you can use an approximate luminosity-mass relation (although you should be aware that stars change their luminosity during their main sequence lifetime and so the luminosity of a main sequence star depends on its mass, age and its initial chemical composition).
If your star is a giant or at some intermediate evolutionary stage (subgiant, or some short-lived supergiant phase) then it is really difficult to estimate the mass of such an object without also knowing its age. For example if you know a giant star is a billion years old, then there is only a small range of masses which will be in the relatively short-lived giant phase at 1 billion years old.