I wouldn't go for the Tunnel de Bielsa. Have been in that region quite a few times in wintertime and it closes down quite often. On french side snow is the problem on Spanish side landslides. If you do go up watch out for stones, mud,.... and sheep using the tunnels as shelter for the rain.
Again, it will be a guess if you get over or not at a high pass. Snow on the Spanish side in wintertime is rare, but it happens. On the French side you are depending on the snowplows.
The tunnel itself used to close at night but I don't know of they still do that.
Also, if Tunnel de Bielsa is not passable the passes nearby to work your way to the coast will most likely also be closed or unpassabel for a bike. That means you have a serious detour to go. And you would loose quite a bit of time.
Mayby checking the internet ahead might help. I am sure there has to be some source that can tell you wich mountain pass is open and wich one is not. Then check the weather for the comming days. And make a plan based on that info.