If your potential is firstly spherically symmetric and secondly your scattered particle does not penetrate too far into the charge distribution then Rutherford's model is adequate. The phrase I used of 'too far' is a matter of accuracy. A Gaussian potential clearly is non zero at large distances and will affect the result.
If you want to consider non spherical symmetrical potentials or when the kinetic energy of the scattered particle is significant with respect to the electrostatic energy of the scattering particle ie it goes 'too far' then you will need to go back to elementary principles - see Goldstein's Classical Mechanics - but be warned there are very few 'closed' ie analytic solutions for such problems.
A help maybe in a totally different area of scattering theory look up the extensions made to Mie's theory of scattering from ellipsoids - the math will help
Hope this helps
Regards and good luck