You could try the spectrophotometric method of determination of the colored product of the reaction of lactate ions with iron(III) chloride at 390 nm. See Article Spectrophotometric determination of lactic acid
This article may help: Taylor, K.A.C.C. A simple colorimetric assay for muramic acid and lactic acid. Appl Biochem Biotechnol 56, 49–58 (1996). https://doi.org/10.1007/BF02787869