Ok so this is my thought process about the whole thing.
You activate your clear turn.
Pay a favor to use Ambush on Bjarrhvit.
You immediately resolve an attack with one net success.
A net success is describe as how many successes you have remaining after subtracting any of the successful defensive dice rolled. No dice are rolled by either party and the end result is 1 net success, resulting in a successful attack.
Is the opponent exhausted?
No-
Perform your 1 success on the damage tree
Yes-
Because it is a successful attack gain an additional success and also a
With your success you elect to push the enemy model.
Will your push cause the enemy model to come in contact with a friendly model?
No-
Resolve the push, follow up, and damage.
Yes-
The first time during an attack you push an enemy model into base contact with a friendly model generate an immediate additional success on the attack.
The wording for the attack states that the difference in successful results between the attackers dice and the defenders dice generates the net successes (if any).
Successful attacks against exhausted opponents add an additional success, it does not add to your net success. Same is said for pushing an enemy into a friendly model, you add a success.