Hi,
Klink actually treats Steam traffic just like any other protocol (e.g., HTTP when browsing or downloading over the web). Instead, I suspect that your problems have something to do with throttling over your network.
To make sure, I just ran a quick test with Klink and Steam. In order to eliminate any issues with the network, I connected my phone over wifi and used Klink with a Windows computer. All other network connections were disabled on the computer to ensure that traffic was indeed going over Klink's connection. Configured in this manner, Klink was basically acting as a USB wifi adapter for my PC. I then attempted to start a game requiring an update. The 750+ MB update download kicked in automatically and completed without a hitch. The download rate according to Steam's client maxed out (but did not exceed) my broadband connection and proceeded without any stalling. Moreover, the download counter in the phone app updated itself as one would expect.
Throttling in the network would explain the behavior that you're seeing. Moreover, it's very possible that Steam traffic is detected and eventually throttled in your situation, since Steam has no analogue on Android. You should be able to test this by using Klink in wifi mode at your friend's place or by using Klink in conjunction with a VPN service (there are some "free" services like Hola and Hotspot Shield that you can test out).
When using an Android app-based VPN, be sure to start the Android VPN first before running Klink. If you run a PC-based VPN client, do the reverse and start Klink first before establishing the VPN connection on your PC.
Also, I wouldn't worry too much about LTE speed exceeding your friend's broadband speed, as that does happen on occasion. Speedtests on an LTE network can sometimes hit 50mbps in ideal conditions! Do keep in mind that data over a phone network is ultimately shared at the tower so performance will vary.
Kevin