I guess I wasn't confused about the concept of algebraic granular temperature, but about how the coefficients from the MFiX documentation relate to the equation from the Fluent theory manual. I finally figured it out - I hadn't realized that you can rearrange and solve the (nonlinear) algebraic granular energy equation using u-substitution and the quadratic formula.

For anyone else who is curious/confused, this is what those MFiX coefficients actually mean:

K1m = coefficient for granular pressure (basically P_s/theta_s). The MFiX documentation ignores the kinetic term of granular pressure and just includes the collisional granular pressure coefficient.

K2m = coefficient for viscous energy generation from velocity divergence "Dmii" (*i.e.* (lambda_s - 2/3*mu_s)/sqrt(theta_s))

K3m = coefficient for viscous energy generation from velocity gradient "Dmij" (*i.e.* mu_s/sqrt(theta_s))

K4m = coefficient for collisional dissipation (*i.e.* gamma_theta/(theta_s)^(3/2))

So basically, K1m through K3m come from the "generation of energy by the solids stress tensor" and K4m comes from "the collisional dissipation of energy" in the Fluent theory manual. I wrote out a derivation but it's rather messy, if anyone reading this post wants some more details feel free to let me know and I can post them!