I could really see a GM reasonably ruling this ether way.
Just to be clear and get the issue spelled out,
Pounce wrote:If the creature reaches its opponent with a leap and the opponent isn’t too much larger, the creature may spend an additional success from the Attack test to force the opponent to make a Knockdown test against a DN equal to the Attack test result
So if the conditions are met, and the Lion spends the extra success, the target must make a knockdown test against the Attack test result, regardless of how much damage was dealt, or even if no damage was dealt past the targets armor. This extra knockdown test is granted by the Pounce ability, so it is not really accurate to say that the Surprise Strike would be an "Second Attack", it would simply be and addition to the damage of a test that is already going to be made anyway.
So I can reasonably see somebody saying that they want to do the Knockdown test before doing damage, and then if the Knockdown test does knock them down, do surprise strike damage.
My own personal interpretation, however, would be that Surprise Strike can only be done if the attacker meets the conditions for a Surprise Strike before the attack is made. So if the target is already surprised, blindsided, or knocked down when the attack is started, then surprise strike can be used, otherwise it can not. If somebody has two attacks and knocks the target down with the first attack, I would allow the 2nd attack to be made with surprise strike, because the target was already knocked down when the 2nd attack was started.
Note also the wording on Surprise Strike,
Surprise Strike wrote:The adept takes advantage of an unaware or distracted opponent to inflict a telling blow. This talent may be used against targets that are Surprised, Blindsided, or Knocked Down. If the adept’s Attack test against such an opponent is successful, he substitutes his Surprise Strike Step for his Strength Step in the Damage test.
The order in which this is written seems to me to say that first you check to see if the surprise strike conditions are met, then you declare that it is a Surprise Strike, then you make the Attack, then you do damage. It just does not seem to me that a target who became knocked down during the attack met the conditions early enough.
But I could easily see other GMs reasonably disagreeing.