My concern with Mosher chemistry is that we have a free carboxylic acid in the molecule, which might react with the Mosher acyl chloride.
HPLC with a polysaccharide based stationary phase or with a chiral ion-pair reagent might work, but both of those are speculation on my part. I am starting to search for examples.
That is IMO not really an issue. Ive run quite a few reactions on free amino acids - be it acylations or various protections and never had issue. You can always
a) separate the byproducts by chromatography b) protect the carboxylate.
If we see only one signal, my interpretation is that this would be consistent with having only one isomer.
That is not entirely true. Your hypothetical diastereomers might have very similar NMR signals but ofc, its quite unlikely that all of them will be same. There is also a (rather unlikely) possibility of full inversion of your acid in the synthesis so you end up with the other enantiomer.
Ive done bunch of work in this area for my PhD i can recommend few papers for you.
[1] J.M. Seco, E. Quiñoá, R. Riguera, The Assignment of Absolute Configuration by NMR, Chem. Rev. 104 (2004) 17–117.
https://doi.org/10.1021/cr000665j.
[1] J.M. Seco, R. Riguera, NMR methods for the assignment of absolute stereochemistry of bioactive compounds, EMagRes 4 (2015) 1–30.
https://doi.org/10.1002/9780470034590.emrstm1398.
Those two will pretty much answer all your questions on the NMR method. IMO making mosher (MTPA) amide or methoxyphenylacetic acid (MPA) amide with your amino acid is the easiest. While MPA is preffered to MTPA, I assume the fact that you can buy MTPA chloride will make the task much easier.
Make amide with both enantiomers of MTPA-chloride, make enough of if to get reliable 1H, 13C and 2D NMR to get full assignment of all signals. Dont forget that S-MTPA-chloride makes R-MTPA-amide due to change of priorities in CIP.