Bladesinger+Buckler

Since a buckler shield is strapped to your arm and not held in the hand a Bladesinger (complete warrior) should be able to use a buckler with his off hand and a rapier/longsword in his main hand and retain use of his class features that require him to have nothing in his off hand, correct?

Bladesinger only requires you not to wield anything in your off-hand. A buckler can't be wielded except as an improvised weapon, and technically a tower shield can't be either. So yeah, a buckler should be fine, but talk to your DM.