So first off I think the Pred should be able to identify weakened targets somehow. Whether that’s through his vision mode (giving us another reason to use it, for the fireteam to mud up, and for them to want to remove it)
I think low health targets should have some kind of visual indicator, or perhaps indicate targets that have been downed once or twice already. Maybe a limp and slight (very slight) movement reduction for low health, then the fireteam needs to protect and even be hobbled by one of their own. And an icon displayed in heat vision mode for targets that have been downed. Maybe a hash mark or Pred glyph above their heads for each down.
Quick claim should only be available on targets that have been downed twice
This would extend matches just a bit. Preds would have to soften targets a bit before moving in. Zerker could still do his melee rush and all that, but he’d either have to wipe the team or hit and run instead of hanging around and burning health for the claim. Fireteam would have longer gameplay sessions and the Predator has more reason for downing targets and even injuring them for movement reductions.
Other gameplay element would probably need to be tweaked to make this work. Some more permanent consequences for the fireteam for being downed. I’d still like to see a health cap penalty or even stamina max reductions for being downed. Just so the Pred can feel like his having an impact even if he doesn’t get a claim. Right now it’s kinda all or nothing. You have to get the kill or downing a player doesn’t really matter all that much. Sure they can be killed on a third down, but with missions being as short as they are it’s more beneficial to get the kill.
Movement reduction for damage would also have the added effect of pushing the fireteam to burn medical supplies more quickly. They’d run out of heals faster. Again, other way the Predator has an impact to just injure or soften his targets. Best part is none of these changes would prevent really skilled players from victory, as the punishments are for taking damage and playing carelessly.
Imagine the fireteam low on supplies, down a man, scrambling to the chopper, 2 covering an injured soldier as he struggles to keep up, the predator stalking the trees above.
Weakening and slowing an injured player gives the Pred more opportunities for targets, while still allowing the fireteam to defend itself.

