Thanks very much for the thoughts and assistance!
I contacted my ISP again and confirmed with them that they are not doing any form of filtering or restriction whatsoever. Not only did the first line tech support demonstrate knowledge of this, but when I spoke to an elevated senior tech to be certain, he himself is an IRC regular (irssi, though
). It would have been wildly out of character for them to be up to no good regarding IRC in 2018 when they are very, very generous on other fronts.
Satisfied with this, I took another peek at the new modem/router, which they admitted to being a little unfamiliar with as it was the first big change in years. Eventually I located DHCP leases expiring every 3600 seconds on the mark as a firmware setting. This, of course, lines up identically with my 6 minute time out disconnections. I simply maxed that option out and it more-or-less "solved" my problem - technically, my DHCP lease will eventually time out, but I have another 68 years, approximately, before that comes to pass
This corrected my issue. For anyone else experiencing this problem, I suggest trying to take a look at the DHCP lease expiration limit in modem/router settings and confirming your new modem/router is not set to some ridiculous nonsense like that. There are no further interruptions to my use of mIRC since changing this value.
Thanks again to the both of you for your suggestions and of course a very sincere thank you to Khaled for making the only piece of software I've been using exclusively since the 90s. No other client has ever been as great as mIRC - best money I ever spent.