Comments to the question (v3):

1. The $X^-$ coordinate has (a part from a zero mode) been integrated out in the light-cone (LC) formalism. The above mentioned LC Hamiltonian cannot fully address questions about the $X^-$ coordinate.

2. To get the well-known expansion of $X^-$ as a sum of zero and oscillator modes including the sought-for $\alpha^-_0$ mode term, one has to go back to the full string formulation prior to the LC reduction, cf. e.g. my Phys.SE answer [here](https://physics.stackexchange.com/a/185297/2451). By doing this, one may show that the $\alpha^-_0$ mode term indeed appears in the expansion of $X^-$ as it should.