Dear Barinder,
Are you familiar with Fumiyuki Momose's "Isogenies of prime degrees over number fields?" If not, you may find it here on NUMDAM In it he performs an analysis of the isogeny character and finds that if $k$ is a quadratic field which is not a class number one imaginary quadratic field there are only finitely many $p$ for which $X_0(p)$ has noncuspidal rational points.