I've use to work for one of Richmond Park's neighbouring Local Authorities. Apologies to the spirited discussion above, but the prospect of an entry fee is remote. Personally, I dont think they should do it even if they could because there would inevitably be exemptions, rich-cunt-factor etc that would maintain the status quo.
As it was seen back then, there are two realistic options. The least effective one, but one that preserves the ability to reach destinations in the park from all gates, is making it one-way. No particularly desirable but would have a marked effect in cutting out at least 50% of the rat running.
The second options, is to place filters on each side of the park so the roads are effectively cul-de-sacs (preserving access to ballet school, car parks etc).
I vote for the latter, but realistically the former option is likely to endurethe least hysteria.
I should also note that the gate at the bottom of Broomfield Hill use to be open to traffic from the A3. One of the old timers in the highways team said it use to be grim and the closure of that gate a in the 80s or 90s made a massive difference (edit: Robin Hood gate circa 2003 and there was a petition against of apparently over 10,000), so there is precedent.