A long solid climb is a steady-state. You will be seated most of the climb.
So the best bar is the one that gives you the most comfortable and upright position so that your lungs can breathe optimally while spinning.
I see there are ramps of 11% gradient and a kilometre of downward slope. So aiming for 6.3% is a bit optimistic. You should aim for a higher grade on average because the downslope skews the numbers.
https://www.bikecalc.com/ and http://bikecalculator.com/ can be handy for trying to figure out your numbers, from a formulaeic point of view. Presuming you can maintain an output of 150W etc you'll take 217 minutes to do 20 km at 10% and it will be at 5.5 km/h
So if we target 6 km/h at 80 RPM then your fixed gear needs 29 tooth chainring with a 49 tooth rear cog.
Bonus is that you get 49 skid patches out of this combination, which is a lot.
These numbers are just examples you need to poke in your own values and see what's possible.
the minimumum chainring size and maximum rear cog size will be hard limits.
Finally also consider you have to come back down the mountain. The climbing ratio will be awful for a fixed-gear descent, so you'll either get a ride down in a car or end up with your feet out of the pedals for the descent leaving you with only one braking solution and poor connection to the bike.
Personally I'd prefer a rear freewheel that can coast, AND to equip two conventional brakes without depending on the chain as a braking solution.
Or walk down the hill (what a waste of a downhill!!)
As a test, I took a local climb at https://www.strava.com/segments/3682303 which is 2.15km long with an elevation gain of 165m and an average grade of 7.3%
My time was 10 min 33 sec, I'm 95 kg on a 12 kg bike.
Fiddling the power number gives me 10.52 minutes at 303 watts at 12.26 km/h
To do that speed on a fixie at 70 RPM (I don't spin very well) I'd need 37 tooth chainring and 27 tooth gear, which gives 27 skid patches.
At 80 RPM I'd need 36 tooth chainring and 30 tooth cog.
Differences here is that it was a tenth as long as your climb, and I was pushing hard - 300 W is well above my FTP. OP will likely be doing between 100 W and 200 W for the whole duration of the climb.