dnd 5e – How does the altered Extra Attack feature of the Bladesinger (Tasha’s Cauldron version) interect with the Eldritch Knight Fighter’s War Magic feature?

The Extra Attack feature of the Bladesinger allows you to replace one of your weapon attacks with casting a cantrip. The Eldritch Knight’s War Magic feature checks if you have “use[d] your action to cast a cantrip.” Technically, you have used your action to cast a cantrip, or at least part of your action. Can an Eldritch Knight 7 / Bladesinger Wizard 6 make a weapon attack and cast a cantrip with Bladesinger’s Extra Attack and then make a weapon attack as a bonus action with Eldritch Knight’s War Magic?