Near the taxi rank, outside the station, is a large area for cycle parking. Obviously this doesn't need access through any ticket barriers, although it does leave the cycle prone to the elements. My opinion is though that it is better there than it was, as the old area near Cafe Rouge was much less pleasant!
I've used both the old parking area and the current one, I can tell you from experience the new one is better. It does have a roof, but it doesn't stretch out too far. It helps protect a cycle, but not as much as other parking areas do. Those used at places likes Evesham and Croy (to name just two I've used) cover cycles much more efficiently.
Here's an alternative with potential, depending on your afternoon plans. Paddington has a nicely undercover cycle parking area, between platforms 10 and 11 (I think those are the right numbers, it's been a while since I was last there), and I am pretty sure there's no need to pass through ticket barriers for it either. This is relying on my memory of a part of Paddington I've never personally used, so I could be wrong, please do check in advance!
Your cycle would be much better protected in there, although of course your plans may make it more challenging to get to/from Euston in time for the sleeper. However, it is worth considering if you have the time to divert to Paddington.
I will be using that cycle area myself in a few weeks' time, as part of a multi-function adventure. More details on that will be forthcoming nearer the time, or at the least in the trip report

If it is any good, I will make mention of it that's for sure.
Final thought, I took my mountain bike with me to Edinburgh on the Down Lowland in October 2020. While it is a modern space, well away from passengers, the cycle spaces on the MK5s are a bit small with narrow corridors to access them.