I would be proactive, and have a plan in place. This may include contacting Southwest (management, PR team, or their existing medical flight team) and finding a real person to speak to about the situation, and what could be done if the call comes through. I'd make it clear up front that you are not asking for a free flight (if you are not), but rather the absolute quickest way to get there, and what that might look like (VIP security / line by pass etc included). Having a plan in place means if the call comes from there, you follow the plan (and have back up contacts in case the first person is on vacation / not on duty etc.)
Same idea for the private jet option, letting them know what the score is, and having the action plan in place if needed and the right people's direct phone numbers would make it a lot quicker and easier than having to start the conversations at go-time.
I'd also look at other airlines with connecting flights too - whilst may be slightly longer travel time, remember that those published flight options give 'legal' connection times, and there may be ways to get a shorter connection in extenuating circumstances.