How to remove hydrostatic pressure from basement floor so we can finish part of the basement?
Do your sump pumps run frequently during wet weather? Does water leak through any cracks in the walls or floor? Do you get puddling or standing water in any area of the basement during heavy rains? Do you have an extremely high water table in the area?

If not, I suspect your problem is normal ground moisture not necessarily drainage water or the water table. The developer that built the house or contractor that poured the floor is responsible for not putting down a vapor barrier. 

I would propose that amongst the other things you are investigating, you have a small section (maybe 1' x 1') of flooring saw cut and removed to look underneath the floor. This could be done in an area that will not be finished and could be a DIY project costing you less than $100. Or hired done for a couple hundred dollars. There should be a layer of heavy plastic under the concrete floor. (You could also leave this open to see if water puddles in the hole.)

If your basement floor does NOT have a vapor barrier under it, I propose all these other things will come to naught. If you are getting moisture leeching through the concrete floor then I don't think perimeter drainage is your problem.

The best solution would then be removing as much of the basement floor as possible and have the concrete replaced with a heavy plastic vapor barrier underneath.

Ultimately, if you finish the basement without entirely solving the moisture issue, I would recommend you never use carpeting. Paint on the basement floor may not hold. Therefore, check into just staining the concrete and leaving it after that if you want some color to the floor. This will still allow it to breathe. Throw rugs or anything else that is absorbent, on the floor will promote mold. You will need a dehumidifier running in most weather except for the very driest weather.

Good luck!