Well, okay, but I don't think it is that unreasonable to think characters with full wizard levels will (and should) drop any pretense at martial combat (the attack action). I know I said "access to 9th level spells" but I really meant something like wizard (or probably sorcerer).
For instance, you bring up Bard. While Bards have been granted access to 9th level spells, their spell list is considerably less offense-related, which is key.
The Bladesinger, in contrast, only loses the school features of an Evoker or Diviner (etc). I don't feel it really is the best chassi to build your vision of a bladesinger.
In my opinion, the design failure of the 5e starts with giving it far too attractive spell options, rather than any relatively minor ability score or armor related issue. Yes, you CAN enter melee as a 5e Bladesinger, but why on earth would you ever?
All the benefits conferred by Bladesinging are excellent... for your survival. But you lose little by staying at a traditional safe distance for a spellcaster. You still have access to excellent ranged cantrips.
THAT is the primary issue that I think you need to adress first.
I'm thinking that for your purposes you need a build that offers less magical firepower, enough so to justify entering melee. You still need "cleric level" of martial ability - I completely agree there.
Which leads us to the Valor Bard. As I see it, it is the basic build that comes closest to your goal in many aspects (reasonable martial features, strong but not exceptional magicks), except possibly the spell selections.
One wild idea I haven't checked out in detail: Have you considered what would happen if you applied the Bladesinger subclass on top of the Bard or Cleric base class?
Zapp
PS. Oh, one more thing - please remember that you don't actually need level 9 spells to become a fearsome caster in 5th edition. This is because your save DC increases automatically with your level (actually your spellcasting stat): a
Fireball or
Hold Person is just as fearsome from somebody with only access to 5th level spells as those who can cast
Wish.
So one simple option might be to create a variant of Eldritch Knight to serve as your Bladesinger. As I see it, you might not have to do more than to give it the full Wizard spell list, perhaps in exchange for a lower hit die, and you're done...?
As a Fighter, you can go with finesse weapons and thus avoid MAD - a Bladesinging Eldritch Knight should be able to get by with Dex and Int, and with Con as its tertiary (but no surprise there)
PPS. I guess the TL;DR: would be "don't focus on the Bladesinger"; the choice to base it on the Wizard chassi makes it too strong on spells - giving it the martial abilities you need makes it almost a gestalt character, which makes it an overpowered special snowflake unless everybody else too gains fat bonuses.