This is more of a one (brain-)cell, brainstorm then a real suggestion. But if one day the Devs would decide to make some PvP alternatives - look no further. The ideas are not really mine, and they are proven to work in other pvp oriented games (I hope I don't need to name them by title).
Assault
Teams are divided into attacking and defending. The attacking team needs to destroy a number of targets (objectives) in a given order (e.g.: target#3 can't be destroyed before target#1 and target#2), while the defending team tries to stop them. There is a time limit, if any objectives remain when the time is up, then the defenders win. If the attacking team destroys all targets - they win.
Optional rule:
If the attackers win, the teams are changed - the attacking team in now defending, but the timer is now set to the time they needed to destroy all objectives (e.g.: team 1 destroyed all targets in 5:17, so team 2 needs o destroy all targets in 5:16 or faster to win).
Bombing Run
Each team has a base, at the center of the map a bomb is spawned. Points are awarded for picking up the bomb and bringing it to the enemy base. A player that picked up the bomb can't use weapons. If the player with the bomb is killed the bomb is dropped.
Capture The Flag
Each team has a base with a flag, points are awarded for taking an enemy flag from their base and bringing it back to your own base, while your flag is in the base. If a player with a flag is killed the flag is dropped, the flag can be picked up and carried away or teleported back to base if touched by a member of the team owning that flag.
Domination
There is a number of control points on the map. A team that takes possession of a control point is awarded a point every X seconds. The team with most points after a given amount of time wins.
Double Domination
There are two control points on a map. The team that takes control of both points for a given amount of time wins.
Warfare (Onslaught)
This one is a bit twisted, but most fun.
There's a number of control points (Nodes) on the map and each team has a Power Core, the goal of the game is to destroy the enemy Core, but a core can be damaged only if there's a chain of nodes in the possession of the attacking team leading from one core to the other. The same goes for the Nodes themselves, you cannot attack unlinked nodes.
E.g.:
/-NodeB1-\
EnemyCore--NodeA-+ +-NodeC--YourCore
\-NodeB2-/
To attack the enemy core you would first have to take over NodeC, then either NodeB1 or NodeB2 then NodeA before you could attack the EnemyCore. You can't attack NodeA before taking control over NodeB1 or NodeB2, you can't attack neither NodeB1 nor NodeB2 before taking over NodeC.