Best you can do for a private transcon flight is $~25K or so. If you buy a Jetsmarter membership you could probably do it for about $10K ( $5K for membership and then $2.5K each for a seat for you and your dog. However things are very up-in-the-air with Jetsmarter right now, so I'd fully research that option before pulling the trigger.
You could also look at a service like
YippeeAir, which will send somebody with your pet on a private or commercial carrier. Could do the Jetsmarter option that way and avoid the membership fee.