Berserker is misleadingly "better" as an attacker, but has other problems
I will start off saying that there are many ways to stat out these two classes and determining the most optimal builds for each class would be overly time consuming and difficult to determine.
Given the following:
Assuming Stats of:
St 18 (Level 4 increase Str by 2)
Fighter Champion 4
- 37HP (+9.5 effective hp from Second Wind)
- 1 Greatsword Attack at 2d6+4 (great weapon fighting style boosts damage a little)
- 13.56 average damage, adjusted for crit chance
- Can take two actions once per rest
- Can heal self for 1d10+4 hp once per rest
- Expect to be in splint armor; AC of 17 (Full Plate soon, in theory)
- Crit on 19-20
Barbarian Berserker 4
- 2 Greatsword Attack at 2d6+6
- 13.65 (27.3 per round) average damage adjusted for crit chance (ever so slightly higher)
- Can give self advantage
- Rage gives resistance to physical Damage
- Expect to be wearing scale; AC of 16
- Exhaustion is a problem
It appears that Berserkers have more HP, resistance, and get double attacks (which is all true), but that doesn't paint the full picture. Not only are you going to be easier to hit with a lower AC (easier still if you attack recklessly), but you're going to be stacking up exhaustion, which you don't have any way to mitigate.
Given your constraint of "only one combat", then the exhaustion doesn't come into play, but if you're berserking more than once per day, you're creeping slowly towards inefficacy and perhaps death, until the cleric can remove the condition.
Additionally, at level 5, the fighter creeps even closer when both classes get Extra Attack; Fighter 2 attacks; Berserker 3 attacks.