Rather than go for 'hit', I'd go with "Whenever the BRV Fighter uses an attack power, he gains THP equal to his CON modifier.". That way, it's once per turn (unless you're using immediate attack powers), doesn't depend on him hitting or missing, and if you use and hit with an Invigorating power, you get extra THP. It also doesn't act like damage reduction as much as the other form does.
The problem with this is that a single minion is still not a threat at all. The Fighter will use an attack power nearly every round and nearly always have full temp hit points. The minion can never get through.
Another solution is to have the normal "when the Fighter gets hit" rule, but add on it that he regains the temp hit points only if he takes real hit point damage.
That way, it handles DP's invigorating issue (which I don't see as a big deal), it allows a single minion to still be a threat, and it nerfs the power without overdoing it.
BRV Fighter: 33 hit points
Gets hit for 6, now has 27 hit points and 4 temp hit points
Gets hit for 3, now has 27 hit points and 1 temp hit point
Gets hit for 5, now has 23 hit points and 4 temp hit points, etc.
Against a minion, BRV Fighter: 33 hit points
Gets hit for 4, now has 29 hit points and 4 temp hit points
Gets hit for 4, now has 29 hit points and 0 temp hit points
Gets hit for 4, now has 25 hit points and 4 temp hit points, etc.