A few thoughts on multiclassing.
I'm not sure I'd rate Paladin sky blue. Yes, Divine Smite and smite spells, but you're moving Charisma from a dump stat to at least a 13.
The stat distribution is horrible yes. But you're taking the focus away from some typical Bladesinger/Wizard playstyles and focusing more on Smite. Quite different from the rest of the guide, and not easy to play, but I think still on the same power level to deserve Sky Blue. I think if I were to ever try this, I would do a three or four way multiclass to achieve something like the Multi-Smite build I touch on in the "One of Many" section.
Ranger (Hunter) 3 feature Hordebreaker, if you are already using the Haste-for-Attack-Action trick you mentioned, can give you another attack as a afree action.
Thanks for pointing this out! Not sure how I missed it. Slightly situational, but clearly the better choice. This extends the Ranger interest beyond what I noted for a level 2 dip. TWF, Hunter's Mark (or maneuvers), and a possible free extra attack, for a three level dip. Compare that to TWF, Action Surge, and Maneuvers for a three level Fighter dip. Definitely a contender.
Sorc has the same Chr issue as Paladin, and brings you a lot less in terms of spells known. But it does have Con saves if you start with it. Twin works nicely with Haste, I wouldn't sideline it. Subtle spell also makes your spells immune to counterspell, so if that's a tactic your DM uses with any regularity it's a great pick.
I've made a few updates to reflect this.
I also have a few specific-to-UA dip thoughts, btu those are only valid if a DM allows UA.
Mystic is the only other INT primary class. You can get a bunch of alternate nifty abilities. If you go Immortal at 3rd you'll get INT temp HPs every round, which will help with your own HP issues. Also the disiplines can help. The focus for Celerity gives you advantage on init, which lets you get your bladesong style up earlier, before people attack you. Psionic Weapon gives something akin to paladin's Divine Smite, but powered by PP instead of spell slots. If this is a blessing or a missed opportunity depends on how quickly your slots run dry.
Warlock (undying light) gets +Chr damage to all fire and radiant, which can add to some SCAG cantrips and a bunch of spells.
Sorcerer (shadow) has Con saves like all other sorcs - but it also gets to make a Con save against getting reduced to 0 HPs. DC 5+damage. You can make the save every time, not just once. Also once you get sorcery points you can use the to cast darkness you can see though, a nice trick to get advantage.
I have referenced some UA content, but not all of it. I'll add it to my list of things to do, but I'm not sure when I'll get around to it.