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.