The first construct race in D&D. Let's take a good look.
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It's possible Crawford may have changed his mind since then, and I've seen no tweets about it, but at D&D Celebration's Ask the Sage Livestream he did say that Bladesinger Extra Attack doesn't change the casting time of whatever cantrip you sub in.
https://m.facebook.com/groups/237333840044573/permalink/1251521025292511/
The SAC similarly mentions that unless an ability expressly states it changes the casting time of a spell it doesn't change the casting time.
So, unfortunately, to my knowledge Mending has no special synergy with Bladesinger.