Bit late now, but I used to use Eurosport player, several years ago. There were issues with bit rates, and, as you've found, you can't record. There used to be a replay function, where you could watch some previous events, but without commentary - which was odd.
I did figure out a way of recording it, by playing on the laptop and using a screen recording program, but it was a massive pain as you had to be there at the right time to start it, so I rarely did it.
I ended up getting YouView through Plusnet, my ISP, with a recording box. For a fiver a month it gives easy to access to all the freeview stuff and catchup (iplayer etc.), plus a handful of "premium" channels including Eurosport 1 and 2 that stream in. They can be scheduled for recording like any other channel. At the time it was the cheapest way of getting a properly recordable Eurosport stream (other than dodgy boxes), and means that in normal years we can record all the live cycling and catch up in the evenings, rather than just watching the one hour highlights programmes.
