Discord has a max limit of 2000 characters per message, an option to add a counter so you know how close you are to the limit would make it easier to send messages since you could then decide how to best split up the message to get past the limit.

Not sure where to put the counter. I don't want to make it take up space alongside the message input. Any ideas?

Best place I can think of is on the status bar on the opposite side of the "User is typing..." message

Ok, that should be fine. It might get temporarily overwritten when some other status message has to be put there (like a disconnect/reconnect message) but shouldn't be a big issue.