Montgomery Multiplication with CRT
The issue is, the message length is now longer than the modulus $p$ and $q$

That's not true.  In the $p$ track, you are raising $M \bmod p$ to the power $d \bmod p-1$; we have $M \bmod p &lt; p$; that is, the value we are exponentiating is less that $p$, as results by Montomery multiplication.  In the same way, the $q$ track also satisfies the requirements on that side, and so everything works.