Consider stabilizing the bottom and then adding the liner Stabilizing commonly is done by adding breaker run rock, with sizes maybe 4 inches max and smaller. You may have to excavate some before doing this, because the end result is you are adding solid volume. Stabilization depth required can be determined by testing one area, with a tapered amount from say one foot to three foot thickness. Then drive on that and use the thickness that this test provided for the whole job base. You may have to waste the part of the treatment area done with the lesser thickness of rock. During the test use what ever compacting devices you plan to use for the whole job.
Chances are that by the time this stabilizing is done, you will have most of the rock voids filled with the soil from below, so hydraulic conductivity of the original soil won't have changed much.
I have used this tapered test method for many other areas needing stabilization, such as pavements. It gives you the most economical thickness needed and you are not guessing.
I doubt that any form of sump drainage alone will work. There also may have to be a drainage layer added before you place the liner, so that your liner won't be affected by excess water from below. Single size aggregate for the drainage layer will work only for a short time before it is plugged up. Instead use a less permeable well graded layer, such as concrete fine aggregate, with perforated pipes at no more spacing than 15 feet. Concrete sand does not plug up. A clean bank-run sand may substitute, but give that a test also before committing to it.