If they don't change the URL, you can create a new show in Replay A/V, paste the URL in it and download it instanly or schedule its say weekly recording.

For this web-site you may use anything. You may download it even with your browser. Although the download can be accelerated with download managers and stream recorders. But some free download managers will be able to download this stream for you on schedule (if the URL is persistent).

You can buy Replay A/V or Replay Media Catcher only if you need some other streams to be recorded!
And you can even buy them both whether separatly or in the Audio Video Streaming Capture Suite

Replay Media Catcher is a unique software that can capture previously unrecordable RTMP streams from Adobe Flash Player. Some of such streams can be found on the following page: Web-sites with streamable Flash video and audio. Do you really need to record them? If so then it is a good purchase.

Replay A/V on the other hand can record streaming audio/video files from Windows Media Player, Real Player, QuickTime Player (HTTP streams only), Adobe Flash Player (HTTP streams only). It can also record from Shoutcast MP3 radio stations automatically splitting and tagging MP3 songs. It can record from other sources as well and it has many pleasant features such as scheduling.

Bottom line: For this stream solely IMHO you don't need any of these programs. In other cases it depends on you what kind of streams you need to record. If you record a lot then Audio Video Streaming Capture Suite might be a good choice.

I don't think there will be any problem installing it on a USB Memory Stick, but I don't think they will work on another PC from it.

Besides frequent file writing (such as writing log files) is not good for any Flash drive unless you don't wanna kill it fast.