Compute minimum phase version of a FIR
In the end I used the Hilbert transform to compute the minimum-phase phase of my FIR.
I have used the following formula:

Where H is the spectrum of the desired filter.
Which has been used in:

Individualisation d’indices acoustiques pour la synthèse
binaurale
Phase Unwrapping for Spherical
Interpolation of Head-Related
Transfer Functions
Transformation characteristics of the external human ear


However I am not sure of the results I got.
A follow up question discussing the results and implementation can be found here: minimum-phase phase via Hilbert transform returned values