This is one of the main reasons I think that all money spent on lawyers should go into an account that both sides get to draw upon equally to pay legal expenses. You simply cannot have a system capable of meting out justice if money can skew "justice" in your favor.
That's an interesting idea. I'll have to think more about what negatives might be involved (simple example: someone badgers you with an expensive to refute claim but uses a cheap lawyer to file the claim, though I guess this is just as possible in the status quo).
The best solution is if free access representation was high quality.
A simple way of doing that might be to give parties the option of their choice of n randomly selected representatives drawn from any practicing lawyer, funded via a fraction of punitive damages redirected to a fund locked to that purpose.