Can You Choose To Beg When You Have Enough Food?
I'm certain you always have the option to beg if you want to. The rules (p.4) say:


  A player who cannot or does not wish to produce the required Food must take a Begging card for each missing Food - players may not give up members of their family to avoid the need to feed them. At the end of the game, players lose 3 points for each begging card.


So your question is really asking what the meaning of "produce" is here. I don't see any evidence that it should mean produce in the sense of "conversion" rather than "hand in the food". Giving the player the choice ("does not wish") would be pretty narrow if you have to hand in food just because it happens to be stored as food tokens (as opposed to, say, raw grain). If you were really trying to utilise the Mendicant under such a rule, you could imagine all sorts of silly shenanigans to try and avoid fish, but keeping grain etc. which are against the flavour of the card.

As you point out, choosing to beg is almost always a terrible idea, but the Mendicant is definitely one place where this is a valid strategy. Even with the discard of two begging cards, the Mendicant is still pretty marginal - although it gives you flexibility, it's only a two food swing (or one food if it's not your first occupation!), after all. The other would be if you felt grabbing starting player or something was so crucial to your strategy that it would compensate for the loss (for example, in turn 13, if it was essential that you renovate a large house before the end of the game).