Treat it like any other component overheating problem. Is there good airflow over the drive, or is it nuzzled at the bottom of your case between two hard drives and a rats nest of cables? Is it mounted on the motherboard back plate?

More info will let us help you more. What’s the temperature of the drive during these large file transfers? What’s the max temp before it powers off and is no longer detected?

You’ll be fine. I’ve lived in a house of 5 (student housing), all pretty big gamers, on a ~68mbps connection (UK broadband sucks). It was completely fine most of the time, with some occasional lag when a couple other people were doing bandwidth heavy things (like torrenting or downloading Steam games).

Perhaps in your router you could set QoS rules to prioritize traffic to your work machines during work hours.