Well, “Web Ticket 24/7” means you can submit the ticket 24/7. The response times are generally in the two- or three-day range, loosely.
As for whether this can be resolved via ticket, or has to be done via live chat or callback, I have no idea. I have seen other threads over the years of people getting their battletag replaced with “Adventurer,” but I am not sure how those problems got resolved.
(See below)
If you check the progress of the ticket you have in, it will list the approximate response time. Note that this is an average estimate, not a definite response time, so it may change as time goes on.
The only advice I can offer for the live chat is to check it at 12 PDT, and if it’s not lit up, check it every fifteen or twenty minutes to see if it is.
Here’s another “adventurer” issue, and it looks like the person took care of it via ticket. Hopefully, you can too!