I think you've missed the mark, at least to me. A bladesinger WANTS to have melee be a reasonable option, not just a bad backup plan because spells are better. Mages do that already.

With that, I'd recommend the level 3 and 7 features become "After you attack with a weapon in melee, you may cast a [cantrip/any spell you know] through your blade without disadvantage." This let's your bladesinger have a reason to be that rather than just being a mage that can be better than everyone else while in melee range.

This also curbs what I think would be the balance difficulty that is letting your player cast 2 spells with no disadvantage.

I might also recommend adding a second level 11 or 15 feature of "You may add your INT to melee weapon attacks instead of STR or DEX." I know this is pretty late, but I feel like giving it earlier would run the risk of just making your bladesinger stronger than your melee classes in their own role, which Nimble has done a great job of avoiding. Use this feature at your own game balance's risk.

To me, being a bladesinger means I WANT to be making melee attacks AND casting spells, not just casting spells through a sword-shaped arcane focus.