I've been contracting for 16 years, had a few perm jobs before that. While it's true that you get paid to do what you're already good at that's the case for everyone.
The difference is, as a contractor you can afford to take a few months off and become good at something else - then you'll get a job doing that instead.
You can also pick and choose your projects more freely and there's usually nothing stopping you having long term projects as well as short term gigs, often working for multiple clients at once.
In the US or elsewhere where health insurance is a thing I would probably go perm.
> Companies invest in training and take a chance on you doing something new inside in the company.
I literally LOL'd at this... If you're great at what you're doing, they're keeping you where you are - and "invest in training" could easily mean a 3 day workshop where you find you already know more than the instructor. No thanks.
In answer to the original question, offer more money than you think you can offer and set it up as a 3 month contract-to-perm situation (don't write that down anywhere if you're in the UK - they'll immediately be inside IR35) so you know it's a good match. Alternatively, skin in the game in the form of shares or whatever - but they'd have to be confident in the business, I've rejected that offer in the past.