Noooooo please don't double. All I can compare it to is like learning to drive for 2 days on a motorbike, 2 days in a car and 2 days in a van, then starting a job as a long distance, multi vehicle delivery driver. Yes you have a licence to drive all of them, but you've been driving for 6 days.
You have limited experience at each level, and per game, so if you started a season as a 5, wth all the other L5 refs, you would instantly be compared to them, in the big pot of refs, be thrown into higher matches, with teams that expect an experienced L5 and play to an experienced ref.
After hearing many refs at L3 - NL talk, they have all saud the sme thing - slow and steady, and take each level as it comes.
I've never met someone who has doubled, and said it was a benefit.
All because you can doesn't mean you should.
If yo do decide to go for it then good luck - think London generally has about 3 / 80 that make it.