I haven't seen a sink plumbed that way before.
If you have a vent somewhere between the kitchen sink and the other plumbing, I would think you would be okay.
In a sense, using the oversized pipe is like having a loop vent without the loop.
I don't know how an inspector would view it there.
I personally wouldn't install one that way unless I had talked to a local inspector first.
And if I did, I would want a vent included like the picture below has.
An island vent looks more like this
The santee on the vent below the floor should have been a combo.