TLDR: In principle, you may not engage in a professional or commercial activity in the receiving State intended to provide you with a personal gain. According to the BOFiP, a private profit-making activity may result in the loss of your status as a diplomatic agent and of the tax privileges associated with that status, subject to activities of a scientific and cultural nature. [S003]
Applicable principle
If you are a diplomatic agent, the Vienna Conventions referred to by the BOFiP prohibit you, in principle, from engaging in a professional or commercial activity in the receiving State for personal gain. [S003]
Engaging in a private profit-making activity may result in the loss of your status as a diplomatic agent and, consequently, of the tax privileges attached to that status. This consequence is formulated by the BOFiP in general terms and falls within the scope of the administrative interpretation cited in that publication. [S003]
Scientific and cultural exception
The BOFiP provides for an exception for activities of a scientific and cultural nature. It therefore does not assert that every activity falling within these categories necessarily results in the loss of diplomatic-agent status and the associated tax privileges. [S003]
However, you cannot infer from this exception a general authorization to engage in a professional or commercial activity. The available information does not define the activities concerned or the applicable conditions. [S003]
Tax scope of the status
The tax privileges associated with diplomatic status are governed by specific rules. The CGI notably provides for an income tax exemption for certain foreign diplomatic agents, subject to reciprocity. [S013]
The mention of a profit-making activity in the tax rules relating to diplomatic agents is not, by itself, sufficient to establish that such an activity is authorized in the receiving State. [S004]
What the sources do not specify
The available sources do not specify whether authorization from the receiving State would permit a person to engage in a professional or commercial activity. Nor do they define any procedure, documents, or deadlines for scientific and cultural activities.
You should therefore retain a general rule of prohibition, with an exception that is mentioned but not defined in the available excerpts.