Just saw this. I am doing essentially the same thing only off a pond instead of a lake. I am sure you have had you lake water tested. I am looking at buying this for getting my water up the hill and to our travel trailer areas:
https://shop.rpssolarpumps.com/collections/livestock/products/rps-400-solar-well-pump-kit
Then uphill the trail (800 ft away and 30 ft vertical) I will dump it at an atmospheric tank (after running it through prefilters, prob 50 mic then a 20 mic):
https://www.plastic-mart.com/product/12917/1200-gallon-plastic-water-tank-tlv01200dg
I don't have anything set in stone after that, but from the tank it will be drawn by a separate solar powered pump system to place it under pressure. That is the point where it will be treated like a house water filtration system on well water and forcing the water through a series of filters and UV treatment. I want the iron and magnesium out of the water (not dangerous but the only two metals not within potable water parameters). One of the filters I am looking at is
https://www.springwellwater.com/product/well-water/whole-house-iron-filter/
But I need to do MUCH MORE research on this final stage and consultation with experts on the final treatment package. Being a licensed engineer, I full well know I don't know shit LOL. The goal is to have house water that is purer than city tap water
For anyone looking at treating surface water, make sure you control the entire drainage basin or the catchment is a large public body (like a lake) so you can be reasonably assured there will be no chemical or biological contamination. I control the entire 30 acres whose runoff feeds this pond.