What would you have done if this trip happened six months from now? Whatever that answer is, that's what you do now. Either not go, drive, or buy another ticket.
Here are some good videos to watch from the National Traffic Safety Board.
http://www.ntsb.gov/safety/children.html
And here's what the FAA says:
Did you know the safest place for your little one during turbulence or an emergency is in a government-approved child restraint system (CRS) or device, not on your lap?
A CRS is a hard-backed child safety seat that is approved by the government for use in both motor vehicles and aircraft.
The FAA has approved one harness-type restraint appropriate for children weighing between 22 and 44 pounds. This type of device provides an alternative to using a hard-backed seat and is approved only for use on aircraft. It is not approved for use in motor vehicles. Learn more about harness-type restraint.
The FAA strongly urges parents and guardians to secure children in an appropriate restraint based on weight and size. Keeping a child in a CRS or device during the flight is the smart and right thing to do.
http://www.faa.gov/passengers/fly_children/crs/
And the AAP:
Occupant protection policies for children younger than 2 years on aircraft are inconsistent with all other national policies on safe transportation. Children younger than 2 years are not required to be restrained or secured on aircraft during takeoff, landing, and conditions of turbulence. They are permitted to be held on the lap of an adult. Preventable injuries and deaths have occurred in children younger than 2 years who were unrestrained in aircraft during survivable crashes and conditions of turbulence. The American Academy of Pediatrics recommends a mandatory federal requirement for restraint use for children on aircraft. The Academy further recommends that parents ensure that a seat is available for all children during aircraft transport and follow current recommendations for restraint use for all children. Physicians play a significant role in counseling families, advocating for public policy mandates, and encouraging technologic research that will improve protection of children in aircraft.
http://pediatrics.aappublications.org/content/108/5/1218.full
Southwest Airlines:
Proper use of a Child Restraint Systems (CRS) enhances child safety onboard the aircraft. For this reason, Southwest Airlines and the Federal Aviation Administration (FAA) strongly recommend that infants and small children who weigh under 40 pounds be secured in an appropriate CRS when traveling by air.
http://www.southwest.com/html/customer-service/family/baby-on-board-pol.html
A bag will keep it clean. It will not protect it from crash forces (the weight of the seat times 150 mph in a runway accident will make it unusable), it will not protect it from being smushed from cargo containers in the hold, and it will not protect it from simply being left behind. It's not a structural items. If it must be checked a box is the only safe way to do it. And as you can imagine, that's not terribly convenient.
Because you're risking the lives of the other passengers on board (weight time speed equals force, and there's nothing stopping your baby on your lap from hitting them) I would strongly encourage you to buy a ticket or reconsider going. I've done both in the past. It's not always the choice I want to make. But it's the safe choice and the one everyone recommends, as you can see. I'm not making this up. You won't find any organization that recommends lap babies.
You've also said she's wiggly. Sit on your couch for however long the flight is, getting up every two hours to walk around your house, use the toilet, and change diapers (if applicable). Then sit back down. Or imagine that trip with her in her own space. Disregarding safety, she doesn't sound like a fun one to hold on your lap for a cross country flight.
The My Ride will fit just fine. I'd put up the armrest for a better fit.
Wendy