As mentioned in other answers, the clear intent of the feat is to improve the single free object interaction you get per turn so that you can sheath or draw two weapons at once - not so that you can both sheath and draw a single weapon in the same turn, effectively getting a second object interaction for free.
…but you probably don’t need it to.
You only need to stow one of your two weapons to allow for the somatic component of your Bladesong cantrip, so you can effectively attack twice and cast the spell each turn, even without the Dual Wielder feat. Though this does depend on your table’s ruling on off-hand attacks, and will require you to alternate the order of your attacks.
On your first turn, you can attack with your rapier and stow it as part of the Attack action, freeing that hand to cast your cantrip as your Bladesong Extra Attack. You can then attack with your whip as a bonus action off-hand attack.
On your next turn, just reverse the order: use the Attack action, cast your cantrip, then draw your rapier and use it for your Extra Attack before using your bonus action to make the offhand attack.
Then you can just go back to the first scenario and keep doing this every two turns.
The reason I say “probably” is the way the Two-Weapon Fighting rules work. The relevant rule is:
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.
A very strict reading of this rule might be that you need to be holding the “different light melee weapon” in the other hand while making the first attack, which would rule out casting the spell in between.
On the other hand, the rules don’t specify that your bonus action attack can’t be made in between your attacks when you have Extra Attack, so if your table allows it, just attack with the rapier, attack with the whip, stow the rapier as part of the Attack action you’re in the middle of, and then cast your cantrip.