The Two Weapon Fighting bonus action attack requires that you take the Attack action, attack with a light melee weapon, and you are holding a light melee weapon in each hand.
The rules for Two Weapon Fighting state:
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
The Bladesinger's Extra Attack feature reads:
You can attack twice, instead of once, whenever you take the Attack action on your turn. Moreover, you can cast one of your cantrips in place of one of those attacks.
So when you do (attack + cantrip), you are taking the attack action and the first attack is with a light melee weapon. Therefore, you would qualify for the bonus action attack from Two Weapon Fighting, if you are holding a light melee weapon in each hand.
This is a problem. As discussed in this Q&A, you must be holding the second weapon when you take the first attack: Does Two-Weapon fighting work if you aren't holding both weapons when you take the Attack action?
This means that for this combo to work, we must be holding two light melee weapons at the beginning of the attack action. Since Somatic components require a free hand, we are now limited to casting a cantrip that does not have Somatic components. Fortunately, there are three such damaging cantrips on the Wizard spell list:
- Lightning Lure
- Mind Sliver
- Sword Burst
Assuming we cast one of the three cantrips above, and are holding two short swords, the following turn is valid:
- Take first attack of Attack action with short sword
- Use second attack of Attack action to cast Sword Burst
- Two Weapon Fighting bonus action attack with short sword.
Booming Blade and Green-Flame Blade
These cantrips will also work, though it takes a bit of analysis to get there. BB and GFB have Somatic and Material Components. As mentioned previously, Performing Somatic components usually requires a free hand:
If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.
But BB and GFB create a special exception for our case here. The rules for Material components state:
A spellcaster must have a hand free to access a spell's material components — or to hold a spellcasting focus — but it can be the same hand that he or she uses to perform somatic components.
The Material component for BB and GFB is:
a melee weapon worth at least 1 sp
Because our short sword is the material component for these spells, a hand that is holding a shortsword can be the hand we use for the Somatic components. Therefore, the following turn is also valid:
- Take first attack of Attack action with short sword
- Use second attack of Attack action to cast Booming Blade
- Two Weapon Fighting bonus action attack with short sword.