I wanted a pond for many years, and am getting close to having one now. Here are my thoughts. I have seen small backyard ponds with a liner, and they are OK. They were generally shallow depressions dug about 2 feet deep in middle and tapered straight flat to edges. The liner stretched and looked smooth. Your dug out pond, ( beautiful by the way, shape ) is irregular. A liner placed into, would be with wrinkles. No liner made, comes with a forever life span. It would need to be replace within ??? years. Now lets look into sodium bentonite option. It is a great product , but may not be the best for your application. ( at least the way normally applied.) It will work great on large flat bottoms. It stays in a paste form and not solid. ( Read this in preventing ponds from leaking from cow hooves entering ponds to drink, etc.) I see 2 drawbacks here. First, it would be a challenge keeping it in place on the vertical walls of your pond. Second, it may prevent you from having nice clear water. The bentonite needs to be held in place with a layer of soil, sand, or whatever else. The way your pond is made suggests that you want to keep water plants and fish. If you were not thinking of fish??? then think again. The fish will naturally control the mosquitoes. Certain minnows are better at eating mosquito larvae than other fish. You experiment there for your best results. For fish, you need clear water, so you can enjoy seeing them, as well as for their well being. A small swimming pool type filtration and a means to aerate are also good for clear view pond water. Filtration does not need to run continuously. Aeration keeps water from going stagnant and developing unpleasant odors. You can do both as you see fit and best results.
Here are my plans on how I will make my pond. Mine will be about half the size of yours due to size of my yard. Not as deep ether. After I am finished digging, I will cover the soil with bentonite, about 2 inches. I will mix up small batches of cement and create a 1 inch thick cement layer. while cement is still not set, I will cover top with decorative river rock pebbles. I don't want the look of a sidewalk underwater. When cement hardens, I will remove all non binding pebbles and keep using them as I keep progressing with more cement troweling. Making cement mix stiff enough, I can apply vertically with limitations. This is more than a single afternoon project. When I have all cement completed, and one day hardened, then I will fill with water to cure cement. ( cement cures well when in water) Other option is to sprinkle with water often for at least the first week. I know there are cement sealers, that can be applied, but I personally will skip those, since I'm planning to keep fish. What I end up with is a thin shelled cement pond with pebbles on top. In time it will develop small cracks of course. That is the reason for the 2 inch layer of bentonite under cement. Water will stay clear not being clouded. The pebbles will keep the cracks out of sight also. It will have a natural look.
Every so often I will have to brush the sides and bottom with a broom to loosen up the green algae. The pool filter will remove all the floating particles as well as the algae.
Hope this gives you some possible ideas for your pond. AND