Here's the actual climb in question, according to OP.
So if you are out of the saddle, clipped into pedals, and have your hips to the stem and are bending forward at the waist, then there's little more you can do to balance the bike.
Consider not wearing a backpack.
You might be able to vary your track on the path by zigzagging, or taking a slightly less head-on approach.
Try the climb again, while recording on strava.
Then use the Compare function, or a link like this https://www.strava.com
/segments/4354748/compare/MzU0NzM3Nzc4NSwxNTgzODc4OTU0NA== to compare your best segment effort to someone else's.
The top time was only doing 1.4 km/h on the really steep bit at 0.7 km into the segment.
Generally the answer is momentum, and lots of it.
And if you stop, you're stuck.
If you're there, try and watch other people ride the same track.
See how they approach the same sections and where they push hard.
bicycles.stackexchange.com/users/19705/Criggie
Don't just lean forward... move forward such that you are sitting on the pointy nose of your seat...
that's right position it
so it feels like you are about to have your rear impaled.
It requires leaning forward
but it also lowers and shifts forward your center of mass.
Here is where I get all nerdy.
Imagine an ellipse whose ends are defined by the front and rear tires contact points with the ground...
the sides (or width) of the ellipse would be a function of speed, terrain surface, and a riders comfort leaning into turns...
so we have this ellipse...
when climbing if a line projected from your center of mass (gravity) to the ground along the earth's gravity vector falls outside your ellipse you are in an unstable position...
So get low, get forward, and keep two wheels on the ground.
Have fun out there.
bicycles.stackexchange.com/users/23679/dafew
I think its the steep turn that is stuffing you up more than the grade alone.
Consider this instead, go around and out along the ridge.
Then turn around on the slight crest, and nail it in a straight line back up the ridge-line.
If you get out of the saddle I bet you won't have a problem.
bicycles.stackexchange.com/users/19705/Criggie
I don't have a section that steep
but I have a front wheel lifting problem on the last part of my commute since I have a lot of weight in the back pannier.
No amount of leaning forwards stopped it when I'm fully loaded.
So I drape a toolbelt over the handle bars which makes a big difference.
The toolbelt weighs around 7 kg.
bicycles.stackexchange.com/users/25176/Kilisi
As well as the suggestions already mentioned there is one tip that may help - when you are on a steep climb you will almost certainly be pulling on the bars - if you are pulling up on the bars then it will obviously encourage the front wheel to lift.
Instead, think about trying to pull backwards on your bars - so imagine that you're trying to pull them towards your knees.
bicycles.stackexchange.com/users/28653/John M
When climbing longer 40-50% gradient sections my front wheel starts lifting even at cadences around 70. Leaning forward doesn't help much, I am already balancing myself on the rear wheel. Today the bike suddenly got out of control, I flipped and fell on my bum.  Longer stem? Saddle forward? Can you shorten the front suspension? What can I do to shift weight forward?
MTB Hardtail preventing flipping backwards on steep gradients
bicycles.stackexchange.com
<hardtail>
bicycles.stackexchange.com/questions/43611
bicycles.stackexchange.com/users/25662/AzulShiva
If you are out of the saddle, clipped into pedals, and have your hips to the stem and are bending forward at the waist, then there's little more you can do to balance the bike.  You could consider not wearing a backpack. Another option to try is not just leaning forward but moving forward such that you are sitting on the pointy nose of your seat. Draping a toolbelt over the handle bars may help move weight forward. Think about trying to pull backwards rather than pulling upwards on the handlebar. Try to shift as much weight forward as possible by moving forward in your seat and bending forward, not wearing a backpack and adding weight to the handlebars. Try watching the technique of other riders on the same section. Reduce gradients by zigzaging and going on the outside of a steep turn. Try to avoid pulling up on the handlebars, and instead try to pull toward your knees.
Try to shift as much weight forward as possible by moving forward in your seat and bending forward, not wearing a backpack and adding weight to the handlebars. Try watching the technique of other riders on the same section. Reduce gradients by zigzaging and going on the outside of a steep turn. Try to avoid pulling up on the handlebars, and instead try to pull toward your knees.