Pumping the free run should help a ton. I would think the strainers would work then.
Didn't mention that, but yes, I always pump the free run out first, so all that's really left to press are the sludgy skins and seeds. By the time the remains in the fermenter are scooped and loaded into the press basket, the balance of the free run drains off pretty well. My practice, before starting to apply pressing force, is to wrap the press basket with saran wrap, which contains all of the little squirting wine streams that may develop as the pressure increases. Keeping up with cleaning the strainers is not hard at all, and makes a pretty clean press run wine.