Why do we subtract the mass of one proton in mass spectrometry (negative mode)?
Interesting question, which originates from a major misconception related to pH in LCMS. Nowadays, there are dozens of ionization techniques so in the question always mention the ionization method.

Anyway, LCMS is usually done with electrospray ionization (ESI) technique and so are amino acids. First of all nobody completely knows the exact process in  electrospray ionization as yet. This is not unusual in mass spectrometry where even a common ionization technique like ESI and electron impact, the picture is not 100% clear.

Second important point is the pH in ESI does not determine whether a negative ion or a positive analyte ion will be formed. I also used to have this misconception a decade ago about pH. The final ionization process in ESI takes place in a gas phase. However it is well known that certain ion can suppress ionization very badly. Don't try to connect mobile phase pH with the state of ionization of the molecule in the gas phase.

In negative ion, you are simply observing a net negatively charged species. Nothing 

For details, I'd encourage you to read "pH Effects on Electrospray Ionization Efficiency" in the Journal of the American Society for Mass Spectrometry· Go to the journal website and locate the paper with this title. I am not providing a link so that students develop the habit of searching themselves.