GHASH with a finite field multiplication algorithm in reverse order
My question is... what's the most effective way to utilize this existant method?


Well, you could rearrange things to take 1 reverseEndianness operation per block, plus 1 reverseEndianness operation per GCM operation (plus 1 per key setup).


  
  Let $X_1$, $X_2$, ... , $X_{m-1}$, $X_m$ denote the unique sequence of blocks such that $X$ = $X_1$ || $X_2$ || ... || $X_{m-1}$
  || $X_m$.
  Let $Y_0$ be the "zero block" $0^{128}$.
  For $i$ = $1, ..., m$, let $Y_i = (Y_{i-1} \oplus \text{ReverseEndianness}(X_i)) \cdot H_{\text{reverse}}$.
  Return $\text{ReverseEndianness}(Y_m)$.
  


where $H_{\text{reverse}}$ is the precomputed $\text{ReverseEndianness}(H)$ value.

This is essentially the same as what you had, except we keep the intermediate $Y_i$ values in bigendian format