Sorry for the late reply, for some reason this time around I accidentally cleared the notifications.
Well, that does sound weird. Maybe they've changed something in firmware to make those games you're trying out to not work unless the update for them is downloaded and installed? And you're right, going into maintenance mode does clear the update nag for the system update, not the game update. What's even weirder is that you created a fresh emuNAND from a sysNAND where everything works apparently, yet update nag keeps coming back for the games.
I'd try cleaning orphaned files via DBI and see if that will solve the issue. Really weird.
Another thing that may be causing the issue is that the games might be incomplete on install? I've had a similar issue with Slay the Princess: The Pristine Cut and Lost in Random, I think. Even though the game says that's it's on e.g. v1.0.1 it still required a v1.0.1 update to be installed separately in order to work.
Tried using JDownloader? That's what I usually use for downloading stuff. As for the error you're getting it's either a browser or network issue. Could be that the name is too long as well. It's been a while since I run into such an issue.
You should definitely give your laptop's HDD a bit more breathing space.

Move some of the stuff to the external HDD and see if that will fix your downloading issue.