Using an isothermal equation of state requires the use of a specialized Riemann solver. Based on the book of Toro, I derived the corresponding equations and implemented an (exact) Riemann solver that solves them. The results obtained for the 1D Sod shock with GIZMO_SPH and EOS_ISOTHERMAL_GAS match those obtained with GADGET2_SPH and EOS_ISOTHERMAL_GAS, and also agree with the analytic solution (which also makes use of the isothermal Riemann solver).
Writing a TRRS isothermal solver is quite straightforward. I still have to look into the HLLC solver and see if it is possible to make an isothermal version of that.
I have a short document in which the equations for the isothermal Riemann problem are derived, and a Python script that can be used to get the analytic solution for a Sod shock type problem with an isothermal eos. If you are interested in these documents, I can also add them to the repository. The isothermal Riemann problem is not solved in Toro, so there is not really a reference for this.