You need to roll well to get any benefits from adding Bladesinger.
Bladesinger is really tough to do. I think if you really want that you dump Constitution, play a human, play a custom background so you can take both the Tough Feat and Magic Initiate Druid and run a 12 Dex:
S13 D12 C8 I17 W8 CH16
Take Magic Initiate Druid-Intelligence with Shilleleagh.
Start Paladin in Chain and shield (AC18) sword and board swinging a staff, get the two-weapon fighting style, which you will not use for a while.
Go 4 levels in Paladin, drop the chain for studded at level 3 (still 18 AC), take warcaster-Intelligence at level 4, then switch Wizard. Go 6 levels in Wizard boosting Intelligence to 20. Switch from studded and shield to mage armor at Wizard 1, which is 1 point worse initially, but you have the shield spell and two weapon fighting. For mastery at P4/W1 switch to Hand Axe and Scimitar. Until extra attack is online fight with a Scimitar and Shilleleagh-Club or Scimitar and Hand Axe in Bladesong and Truestrike with a Shillelagh-Club when not in Bladesong.
After 10th level I probably go back to Paladin, pick up 11 levels in Paladin (boosting Charisma at P8), then one more Wizard level and take your last two levels in Wizard 8 and Paladin 12 to pick up two epic boons
At level 20 your stats should be: S13 D12 C8 I22 W8 CH18 with 128 hit points. AC is 18 or 24 in Bladesong before any magic item or spell boosts. Saves are good S+5 D+5 C+3 I+10 W+9 CH+14
I think this character struggles a bit in levels 5-10 until the Bladesinger extra attack comes online, but using Truestrike and Shillelagh helps. I don't think this is better than a single class Dex Paladin though. If you can't do a custom background then you can't get both Magic Initiate(Druid) and Tough. This makes it quite a bit worse at low levels. You probably want Merchant for Lucky if you have to go this way.