Basically, add in a bonus score multiplier that is rewarded for living with a killstreak of like 10 kills in a life (this should be scaled based on the gun being used), and for doing Teamplay based actions like reviving, transporting, capturing objectives, etc. The bonuses should stack so you can earn as many bonuses as you're able to!
It could be something like adding 20% score to every thing that gives you score. So a score message might say "revive 100 + 20 bonus". Ideally if you stack a few bonuses you could be earning a bunch at a time (so it would be like "revive 100 + 60" with 3 bonuses).
The bonuses could give different amounts based on what the task is, or all be the same imo. Either would be good. Perhaps give an extra 10% (30% total) for bonuses that pertain to a class's role. Like medics getting 30% for healing while everyone else gets 20%.
The bonuses would NOT go away when revived. Don't want people getting upset about getting revived (and don't want suicide c4 being punished more). Also when people beg to be revived in deadchat they'll be more serious 😌.
You could also show on the scoreboard a little icon if someone has a bunch of bonuses to reward them with bragging rights. Of course this should also be put in on the leaderboards at the end of a match.
I think this would incentivize staying alive pretty effectively for most players while also rewarding people for Teamplay and such.