I think you should centrifugate your medium and perform a transesterification on the liquid fraction, also taking into account the experimental error brought by the dead cells that are certainly present in medium.
Centrifuge the cells away, extract the supernatant in solvent (ethylacetate or similar), dry unter nitrogen stream and then analyze the fatty acids as methyl esters by GC or GC/MS. the analytical method is routine to quantify fatty acids from microbes and there are routine protocols available. the resolution power of the GC column hereby allows to fully separate fatty acids by carbon atoms and saturation degree.