I have a Paladin with Dueling fighting style who uses a Longsword but the character sheet only calculates the base +2 STR +2 Proficiency without adding the additional +2 for the fighting style.

Is there something I'm doing wrong? how can I get the character sheet to show the appropriate modifier?

From the rules, emphasis added:

Dueling

 When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.

In the image you, show, the dueling fighting style bonus is being added to the damage for one-handed use (proficiency bonus, only applies to to-hit, not to damage).

Looks fine to me. The To Hit is +4 (2 from Strength 2 from Proficiency). The one-handed damage is +4 (2 from Strength 2 from Dueling Fighting Style). The two-handed (versatile) damage is +2 (2 from Strength only). They are all correct.

As ArwensDaughter said - you don't add proficiency bonuses to the damage rolls - only the to hit rolls.