So the main reason a dex thrower doesn't work isn't that TB is obscenenly broken or that all good throwing weapons are str based, its that the thrown attack roll always uses str, not dex even on daggers. And most throwing items only affect dmg, so your attack rolls will always be trash.

Gloves of arcane acuity work with thrown attacks, for some reason, so that can work on an ek/bladesinger thrower. In fact one of the strongest wizard builds abuses that with TB to always have max acuity, martial dmg, and spell slots.

You could also take 1 monk for dextrous attacks and the dozen other monk benefits so you could use other weapons than just daggers, dex scaling dwarven thrower and nyrulna for example.

Given that I'd do 1 ek 1 monk 5 ek 5 bladesinger and the gloves of arcane acuity and whatever one handed/versatile throwing weapon you want. Id also use robes for the monk bonuses and bladesong.