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That works. When the Champion ability resolves, you just choose not to exile anything, and thus have to sacrifice the Changeling Berserker . It did enter the battlefield, so Reaper King 's ability triggered.

The reminder text on most creatures with champion would answer this pretty clearly, but let's bring along the Comprehensive Rules as well:

702.70a Champion represents two triggered abilities. “Champion an [object]” means “When this
permanent enters the battlefield, sacrifice it unless you exile another [object] you control” and
“When this permanent leaves the battlefield, return the exiled card to the battlefield under its
owner’s control.”

 The first champion ability cannot, and does not trigger until the creature with it has entered the battlefield.  So, yes, your Changeling Berserker will cause Reaper King 's triggered ability to trigger, even if you choose not to sacrifice a creature.

Edit - Also, besides the fact that it says that it has entered the battlefield, further proof that it's on the battlefield:

701.14a To sacrifice a permanent, its controller moves it from the battlefield directly to its owner’s
graveyard. A player can’t sacrifice something that isn’t a permanent, or something that’s a
permanent he or she doesn’t control. Sacrificing a permanent doesn’t destroy it, so regeneration
or other effects that replace destruction can’t affect this action.

The fact that Changeling Berserker can even be sacrificed is proof enough that it has indeed entered the battlefield, as it wouldn't be a permanent (and would not be able to be sacrificed) otherwise.

Thanks everybody, I thought that's how it went but I just needed somebody to back me up.

So there was this other weird thing that happened during that game that we were both unsure about, so here it goes:
He had in play butcher of Malakir and Grimgrin, Corpse-Born . I had in Play Reaper King , Lightning Greaves (attached to RK) and Changeling Berserker . Berserker had championed Adaptive Automaton  when it came into play.

It was his turn and in his attack phase he declared that  Grimgrin, Corpse-Born was attaching, and Grimgrin's ability was targeting my   Changeling Berserker . So what happens first? Does my  Adaptive Automaton come into play before  Butcher of Malakir ability (triggered by CB entering the graveyard) causes me to sacrifice  Reaper King ? Do I get to blow something up with Reaper King's triggered ability, or does he get sacrificed before the Automation comes into play.

Wait -- in the scenario as you presented, Butcher of Malakir 's ability hasn't triggered at all.  Its trigger condition is "Whenever Butcher of Malakir or a creature you control is put into a graveyard from the battlefield."

Since your opponent controls Butcher of Malakir , and you are sacrificing Changeling Berserker , the Butcher's ability doesn't trigger.

Edit - Let's say your opponent controlled a fictional creature whose trigger condition is "Whenever a creature is put into a graveyard from the battlefield, each opponent sacrifices a creature."  We'll call that fictional creature "Thing A" for the following example.  In that case, it would play out like this:

Changeling Berserker and  Thing A's triggered abilities both want to go on the stack at the same time.  To handle situations like this, APNAP (Active Player, Non Active Player) order is used.  That is, the active player puts all of their triggered abilities on the stack first, and then the non-active player does the same.  Since it is your opponent's turn when this is happening, this means that your Berserker's ability goes on the stack after Thing A's ability.  Since it was the last object added, this means that you get your exiled creature back in time to sacrifice it for Thing A's ability.*

* Again, though, this scenario is based on a card that doesn't exist, and is merely handling how things would work, if such a card existed.