How does the chemical potential allow for open systems to be considered?
Actually, there is a completely valid definition of chemical potential in terms of enthalpy (I realize your confusion on this point may be my fault, since I initially told you the opposite on another post.)

The correct form for chemical potential as a partial molar enthalpy is (as I think you already suspected):

$$
\mu_i=(\frac {\partial H}{\partial n_i})_{p,S,n_{j\neq i}}
$$

So the total exact differential is expressible as you had originally expected:

$$
dH=TdS+VdP+\sum_i\mu_idn_i=(\frac {\partial H}{\partial S})_{p,n_i}dS+(\frac {\partial H}{\partial p})_{S,n_i}dS+\sum_i(\frac {\partial H}{\partial n_i})_{p,S,n_{j\neq i}}dn_i
$$
  The definition of chemical potential as a partial molar Gibbs free energy is only valid at constant temperature and pressure, so it wouldn't make any sense to write it that way in the expression for the exact differential of enthalpy.

Sorry for any confusion my initial errors may have caused you.  Hope this clears it up.