As an interim solution since you said you can't shut down the system, at least maybe for a short time...
Remember that the data type is a signed INT (pos2bil to neg2bil), meaning there are 4bil records allowed, not just 2bil. So, why not consider doing a DBCC CHECKIDENT to change the identity to increment from negative nbrs, and let it go for another 2bil records until you figure out what to do. That should be a quick fix.
Hopefully, you shouldn't be using the IDENT for any business purpose other than just FK, like order by, to care about the order of the records on disk if IDENT is CLUST.
So careful about certain considerations of use no negative nbrs.... But you're in a hard situation that this may work for you, so I'm just providing an idea for a way out...
This is one of the things I don't like about signed data types for IDENT, "loss" of 2bil entries.