I am sorry but Katara is by no means Zuko's equal. She is terrified of Tai Lee because without her bending she is nothing. You can take away bending from Zuko and he still has a whole truck load to fall back on. His skill with broadswords is so far unbeaten by any mundane opponent and only matched by jet who managed to with weapons take on and beat at least temporarily the avatar. He is in fact a genuine Ninja and has the practical skill of inflitration enough to TWICE infiltrate flawlessly a stronghold and go so far as to get most of the way out WITH a prisoner in tow the second time, escaping undetected the first time.
His skill set expands beyond bending. Where as Katara is tied intrensically to her skills as a bender.
First off lets examine the incredibly subjective power of water bending.
There are five phases of the moon, Full, Gibbous, Crescent, Half and New. Fire benders lose their powers in a solar eclipse, and water benders lose it during a lunar eclipse, fire benders lose and gain power with the rise and fall of the sun, water benders do the same but with the moon
So lets put some imaginary numbers to this.
So assuming that it is the moon not the night that water benders draw power from, then we can say that.
((Day = New Moon)) for water bender
Furthermore if the additional power is correspondent to the other 4 phases of the moon, the water bender spends at least 50% of the nights and all of the days with less than half moon power, whatever that means. So Lunar Eclipse = NO power
New Moon/Day = Minimum power
Crecent Moon = Some power
Half Moon = Half power
Gibbous Moon = Great power
Full Moon = maximum power
Now even of these are only averages a firebender is far simpler
Solar Eclipse = NO power
Day= More power
Night = Less power
Certainly the shift is not as dramatic as with water benders as there is no gradation to the phases of the sun, it just is. So if Katara w/full moon, Zuko w/o Sun = Contest w/katara winning
And Katara w/o moon, Zuko w/sun = spank Katara
then Katara w/full moon, Zuko w/o sun +fatigue of hauling aang through a blizzard, and swimming in artic waters = Spank zuko
then we can infer one thing. 95% of the time, IE all days more than half nights a well rested Zuko is going to pimp slap Katara. Furthermore since he has yet to use a weapon in conjuction with his bending, which she has no skill there is even further doubt in my mind that she has a chance. Katara would need the gibbous moon and greater to stand a chance and if Zuko brought his broad swords then the only thing that would save her would be a full moon.