I would rather the solution be continued self-regulation but with better defined parameters and, most importantly, in the full view of the public.
External and 'independent' regulation always ends up being influenced by whoever is in power. The regulators are almost always at worst cronies and at best 'reliable'. And on a larger canvas, it's another tiny cut in the slow but seemingly certain death of the practice of self-government; a practice that elsewhere constantly proves itself to be both effective and supportive of some virtue (in almost every aspect of the Web, for instance).
And anyway why on earth should MPs be let off the hook? They'll be even more inclined to adopt the 'I was only following orders' defence. Let them sort it out themselves and let the public be their judge. It would - and should - be an example to us all.