You can make attacks with a cantrip from that space: but other features of the cantrip may not work
First of all, great question! It's a very clever reading of the rules, and an interesting edge case. As you stated, the rules for an echo knight state:
When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo’s space.
Now, imagine that your bladesinger/echo knight character was to use the Attack Action, and replace one of their two attacks with Booming Blade. They would then need to make a weapon attack as part of that spell. Some DMs might question whether this counts as taking an "attack you made with that [attack] action," but personally, I'd consider it well within a reasonable interpretation of the rules. And since (as you said) the Echo Knight's rules state:
When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo’s space...
you would be free to have the attack made as part of Booming Blade's casting come from your echo's space.
However... note that this is different than changing the cantrip's range or point of origin. Neither of these has been altered. Booming Blade has the following feature:
Range: Self (5-foot radius)
If the creature you are attacking is more than 5 feet away from your character (your actual character, not their echo), then they are outside of range for the spell. They are still in range for the weapon attack you are making as part of casting the spell, because the specific rules of the Echo Knight's attacks override general rules about the range of your attacks. But other features of the spell (like the extra thunder damage) may will not work, since your spell's target (the creature hit with your melee attack) is not within the spell's area of effect (which the echo knight's features has not changed).
The thing that makes this line of reasoning difficult is that other spells, like firebolt, have attacks which can't reasonably be said to do anything if the target is ruled to be outside the spell's range, even though all they are is an attack. If firebolt's attack originates from the echo's location, then what does that even mean if the spell's range and point of origin are unchanged? It's a muddle that complicates more than it clarifies.
A DM will need to adjudicate this. Personally, I'd allow all the effects of Booming Blade to apply to the target of your attack, as I would allow you to cast firebolt from your echo's location if done as part of your Attack Action. After all, this is a build that requires 9 levels of (in many ways suboptimal) multiclassing to accomplish, so it makes sense to reward the synergies that it does have. But the question is sufficiently open ended that a DM will need to make a final call.