Since you already said there is no chance for new servers or expansion of current hardware. Can you modify the software token app or companion app to hold queue for log in?

Let us log in and queue up in the app. Takes load off live servers. Shouldnt get a 2002 error while app is holding position. When queue pops, send alert, give 10-15min to log in or lose place in line. No one has to watch their pc screen for 4 hours this way.

Similar to the queue holder devices used in restaurants and amusement parks.

Just a theory, idk how long it would take to develop and implement, but should be way cheaper than new servers and on a faster time table.