I think it's fairly likely Vincent would be abbreviated to Vinnie. You may be successful with Vince but if so I'm sure there will be people, very possibly including yourselves who are unable to resist Vincey when he is little. But you know, that is okay and Vincey could become Vince as he grows older.
I don't love it but it is a perfectly usable name. It gained a bit of traction as a gangster type name as it featured for those kinds of characters in popular culture a fair bit - I think that's where you're getting the chav associations from. But I wouldn't say that defines the name. It is definitely also a classic, and like another PP I can see a softer/more feminine/artistic interpretation of the name. It brings to mind Vincent from Gattaca, one of my favourite films!
Arthur is a great name, very solid and I'm a real fan of Art / Arty (though you possibly wouldn't be?). It's certainly not too old fashioned, I'd say it's a currently fashionable/trendy name, fairly commonplace but not overused, at least where I live.
My suggestion would be Lawrence. It's a strong, historical but very underused name with the nickname Lance, which you might like if you like Vince. I'm sure you could avoid Larry as that is so dated now, but Laurie might crop up which is rather feminine so that might put you off.
I think Alexander is another good suggestion certainly for a strong, conquering name! And you have a wide selection of less conventional shorter masculine nicknames available like Alec, Lex, Xand if you like that sort of sound.
You could also consider Maxim, just throwing out a wild card.