Bikes on French trains

Hi ! I've done the Lyon-Paris many times as I'm french.
With your bike 2 options:

• TER (slow and regional trains) 5hours, less expensive, bike-friendly areas inside (at least, it should, but anyway its okay to have your bike non-packed).
  
• TGV (fast trains) 1h50-2hrs. Bike shall be packed but you can use almost anything as long as it's sealed and don't brake. Proper bike housing is nice, but a tarp or a heavy duty plastic garbage bag works fine to. You want your luggage to look "as professional" as you can, but don't bother with a big smile and not so many people in the train it should be ok.
  
  

Thus, maybe, avoid train with much affluence (early / late)