Team A is at the line shooting a 1-1.
Team B member steps into the lane early.
I give the "Delayed lane violation" signal.
Ball bounces around the rim and falls off.
My partner blows the whistle.
We get together and he has a pushing foul on the defense.

Not that this is a hard one to call.....but.... I have never seen it before.

We cleared the lane and let Team A shoot the 1-1 and then we went the other way and let team B shoot a 1-1 with players along the line.

Was the pushing foul actually on the offense (Team A)?

If what you posted is the order in which things happened, then you may have erred.

AS soon as the ball "falls off" the ring, it's dead. Contact during a dead ball (the push) should have been ignored (since it wasn't I or F).

(I recognize that the foul may have occurred before the whistle -- so you may have been correct, but I'd look for a way to ignore the foul.)

If the foul was on the defense (B), why did they get to go shoot free throws?

The pushing foul was on the offense.....my bad....