May one shave during three weeks for a 1st or 2nd shidduch date? (since for example looking unkempt can lead to a financial loss and there are hetarim for that, maybe this is similar).
One can be lenient in this matter (shaving for a shidduch), if one thinks that coming unshaven will make a bad impression.
At the same time, the girl in question will understand why you didn’t shave, and there might therefore not be a need for it (on the contrary — you’ll have to explain why you’re clean shaven) — in which case you shouldn’t shave.
The prohibition against shaving in the Three Weeks is derived from the customary prohibition of not taking haircuts.
However, it is noteworthy that the Magen Avraham (551:14) writes that we do not take a haircut in honor of Shabbos during the Three Weeks because it is not the norm to take a haircut each week. This implies that where it is the norm, there’s no problem to do so, and this will apply to shaving (which is done every day or at least every week).
See also the ruling of the Peri Megadim (551, E.A. 14, citing Magen Avraham), and see Biur Halachah to Se’if 3.
In spite of this, the general custom is to refrain from shaving during the Three Weeks, and several authorities rule stringently on the matter (see Kaf Ha-Chaim 551:66). However, as the question notes there are authorities who permit shaving where there is a possible loss of income (see Kaf Ha-Chaim 493:19; Iggros Moshe, Orach Chaim 4:102), and there is room to compare the matter of a shidduch to a potential loss of income.
Therefore, if one thinks that coming unshaven might cause a negative impression, one can be lenient – but as noted in the answer, it is not clear that coming unshaven in the three weeks will cause a bad impression, and this depends on personal circumstances.