The intent is that everything on the Strut goes off as soon as it hits the Ring, whether you played it from Backlash or just put it into your Ring. So let's make that clearer with some errata.
The Stylin' Strut
Mid-match Action
This card is considered non-Unique for The Best Thing Going Today.
When your Fortitude Rating is 15 or more and this card is in your Backlash, after successfully playing a maneuver and damage is resolved, you may put this card into your Ring.
When this card enters your Ring, your opponent may discard 3 reversals; if he does, your turn ends. If he does not, you may draw 3 cards or search your Arsenal or Ringside for a card with 'whoo' in the title, reveal it, put it into your hand, and shuffle your Arsenal; and the next card played this turn is considered to have 'chop' in the title.
F: 0 D: 0 Unique
V1
This errata takes effect immediately.