One solution to sediment-laden water is a settling pond.
If you fill it from your gate, then MOST of the sediment should be settled out within a day. (Charged clay particles can stay in suspense for a long time, but some are small enough to go through irrigation drip emitters without clogging.)
The best practice is to then place your water inlet pipe a few feet below the surface of the water. That way you do not draw sediment off of the bottom, or draw in leaves, twigs, or pine needles that are floating on the surface.
Most of us try to avoid having trees immediately adjacent to the pond. Their roots will try to poke through your "bentonite seal" and the leaves will slowly accumulate in your pond and leave it a little shallower every year. Further, having a "sealed pond" that allows for seepage to water trees is going to be a very delicate balancing act. I don't even know how to make suggestions for that?
If you are not going to use gravity to feed your water system, there are lots of little 12V pumps available. You just need a deep-cycle battery and a small solar panel to charge it. Put the pump on a timer, and then pump for one hour a day (for example) into the adjacent irrigation system. That way you could plant your trees in a line away from the pond, in a circle, or in any pattern that looks nice for your farm. Your sunlight to run the pump will be at a minimum in November-December-January. However, I think that is your rainy season?
Your bentonite MIGHT work without a roller. After you spread and mix it with the soil in your pond bottom and sides, I would bring up a load of water and then mist your mixture so it is inclined to stay in place. You will then need to SLOWLY fill the pond. A big rush of water all at once will tend to move your bentonite mixture out of the way and perhaps leave a path for a water leak.
Most people in the U.S. probably use a grass type matched to their region to serve as their erosion control on the banks of the newly constructed pond. Some people even use a blend of grasses. (Typically with something that comes up fast like rye to jump start the process.) I would select something functional AND attractive, since the area around a pond on the hillside will probably be visible from many places on your property. I suppose you could even do a mix of native wild flowers if you have something that has a dense root system and likes to spread.
I would try to arrange to water your planted vegetation at the very start, so you can get it well established before it has to survive through the next dry season.
We bought 400kg of bentonite clay, but the ponds are too small for a roller, and we do want some seepage, not completely sealed up. How would you recommend using the clay to seal without a big machine?
There is a very specific process for doing that correctly. I am no kind of expert on that topic.
You should probably write that up as a new post in the "Soil Questions" sub-forum. Add any information you can about your existing soil, and the area and the depth of the ponds you are planning on sealing. Hopefully, you can get some specific expert advice on that topic.
