I have played a fair amount of online and offline TDU2, but I have never been able to switch from offline to online mode in-game. When consulting the manual or googling all I can find is something about creating an offline account, nothing about if there is a way to sign on in-game. Do I really have to quit the game and start it again?
I believe so.
As an aside, people who created an offline account originally had to create a different online account so that they could play the game online and they could switch from offline to online by switching profiles in game (From the profile tab). This might be an option for you but I'd be aware that it doesn't sync your saves from offline->online when switching (As they are two different profiles). But you probably know this already.
I have done a Google myself and even though the post is old it seems that you are not alone with the problem that if your online account is used for offline play that you can't go online:
I think it might be related to the number of cheats that you can apply while offline. I'm not clear how the online circumvents this or why it would matter that you can't do that in game but it seems this is their rule.
I know it's not the best, but I hope this helps.