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No reason not to get a 5/600 (or even a 300 in todays market) over a 125. A few good reasons not to go bigger than that for a first bike. Insurance would be a good one.
I would leave aside any ideas of getting a cool first bike and just go for the standard 5/600 class. Something comfortable for your physique and fitting the commuting you want to do.
If you are starting at this time of year you'll need a winter jacket and trousers, there are plenty around that will get you through even if you get a bit colder than you would in the expensive stuff. I started out with a Hein Gericke Master jacket that I wore out over 5 years, it was a budget option but stood up well. Literally stood up on it's own after 5 years of ingrained dirt.
I think I'll wait till I've done the CBT to decide but I'm leaning towards direct access, just got the thumbs up from Mrs Hammer to use a week of holiday for it... Now just need to rustle up the cash.
Is there any reason not to get a bigger bike other than the temptation to nail it and consequently run out of skill?