My recollection is that in one of the last chapters of the book by Denker-Grillenberger-Sigmund, there is a theorem which says something like: given $n$ ergodic measure-preserving transformations with entropy strictly less than $\log d$, there is a minimal subshift of the full shift on $d$ symbols which has precisely these invariant measures. Going to the Springer website, I think it's chapter 31, but unfortunately I can't get through the paywall, so I could be quite wrong...

My recollection is that in one of the last chapters of the book by Denker-Grillenberger-Sigmund, there is a theorem which says something like: given $n$ ergodic measure-preserving transformations with entropy strictly less than $\log d$, there is a minimal subshift of the full shift on $d$ symbols which has precisely these invariant measures. This would certainly be fine if $n=2$ and two systems with identical entropy are given. Going to the Springer website, I think it's chapter 31, but unfortunately I can't get through the paywall, so I could be quite wrong...

That said, if it's not important that the topological entropy of the subshift is positive, then all that's needed is an example of a minimal subshift with zero topological entropy which is not uniquely ergodic. An example of this appears to be constructed in Section 3 of the 1984 paper "Toeplitz minimal flows which are not uniquely ergodic" by Susan Williams.