I think that your assertion that the dove is a sympathetic symbol is an assumption. Dante draws from plenty of classical sources; in classical myth the dove is associated with Venus (eros as opposed to the caritas of the Holy Spirit). Note also that Dante is careful to say that Francesca and Paolo come from "the flock where Dido is" (85; Musa's trans.). Dante wants us to recall Book IV of the Aeneid here.
That said, the idea that the dove is part of a series of images and words that convey sympathy certainly fits in with the psychology of the entire episode. In telling the first part of her story Francesca uses the language of the courtly love tradition to attempt to excuse her sin (while what she actually says reveals that she first praises her own beauty and ends by wishing her killer be buried in hell.) Her graceful speech has the intended effect: it leads to the Pilgrim becoming dazed by (false) pity. She then readily agrees to his request to give the precise details of what led her and Paolo into giving into lust. It is enough to make him faint and fall to the ground, symbolically becoming one of the damned.
After all, the lustful are higher up in the second circle precisely because theirs in a common, "warm-hearted" sin of indulgence as opposed to sins such as treachery. While Dante is honest about how easy it is to fall into lust, the end of the episode--with the Pilgrim on hell's floor like a dead body--clearly shows that the two lovers deserve their condemnation.