Oh by the way so you don't make a mistake about this in the future, papyrus's training from undyne is learning how to cook, not self defense or anything like dat
Have you seen the way Undyne cooks??
On a more serious note, Undyne respects Paps as a fighter but knows his nature and never wants her friend to be in a situation where he might be forced to take a life or die. As you know from the genocide route Papyrus doesn’t survive those kinds of things. Papyrus is a hard worker and a hard-core believer. Undyne may have been trying to convince him to pursue a different dream but I don’t think that would have stopped Papyrus from learning from her. Keeping yourself sharp and in shape is useful even if you don’t learn new techniques every week.