Pete,
I do know that American Airlines specifically says "Bicycle is in a hard-sided case, bicycle bag or box built for bicycle transport". The question for me is a mattress bag a "bicycle bag" and does "built for bicycle transport" apply to the box only or to the case, bag, and box.
United Airlines doesn't mention a box at all. They have changed their requirements because I used to be able to just heavily wrap the bike with HVAC insulation tubing and they accepted it. It was never damaged.
Delta's policy states "If the bicycle is packaged in a soft sided travel bag or anything other than a hard shell case, a limited release form will need to be signed" which to me implies to me that you could pack it in a durable plastic bag. However, their $200-$300 fee each way for over 80 linear inches makes me avoid Delta.
Southwest Airlines states "Bicycles packaged in cardboard or soft-sided cases will be transported only with a limited release of liability. Customers are not entitled to reimbursement or compensation for damage in situations where a limited release of liability applies." so I guess you could do a mattress bag also. I have found SWA to be the roughest of them all but that just may be because I used to fly them a lot. Plus they don't reimburse for damages unless excessive damage to the box.
I have always been jealous of how Europeans can just use a soft but durable plastic bike bag to transport the bikes. Plus the baggage handlers seem to give a better care there when I have observed them moving someone's bike.
Tailwinds, John