I know little about fantasy grounds, but I don't think the CPU load or the computer speed is a factor at all. If you detected that Fantasy Grounds used 100% CPU, or close to that you might start to consider if there was a bug in software that made it freeze, but that does not seem the case.
Most likely I think it is some kind of connection problem. It is often the case that data transmission over an active network socket is something that happens in background, but that new sockets is a blocking operation that prevent the rest of the application to get work done.
If the server software or some of clients is improperly configured you can end with the situation that the clients halt waiting for a socket to be formed, but that the request is intercepted by a firewall or the request is made into nothing so that the connection is never created. If unlucky the combined set of timeouts that determine how long each attempt should take can take many seconds or even minutes. There are multiple timeouts to configure so it is easy to mess up.
That the software eventually recovers could be explained by the Fantasy Grounds software testing different access methods until it find one that works. In most situations this would be considered a bug in the software. The reasonable way to program the software is to test each method with a very brief timeout, and only falling back on the long timeout periods if all methods fail. The absolute best would of course be if the client software itself could diagnose the problem and inform the user that some kind of firewall is preventing proper functioning. For instance I could imagine that the Windows firewall accepts outgoing connections for Fantasy Grounds and incoming connections from Fantasy Grounds for the same server that you connect with, but that it blocks incoming connections from other servers used by Fantasy Grounds.
With all this said I think it is also possible that HP Digital Image monitor is using the same port as Fantasy Grounds and this causing the intended traffic to the FG client never reaching it.
I think the first thing you should do is to determine what firewalls you are using (they might be included in your anti-virus software) and then search on the internet if there are reports of other persons having similar problems with Fantasy Grounds and your firewalls. Secondly you do the same type of search with "Hp Digital Image monitor" included and see if you find any such descriptions. Most likely your problem has nothing to do with the RM parts of the Fantasy Grounds software, so don't include RM or Rolemaster in the seach.