Regarding using "fragile" tape... We had it all over some antique clocks we were bringing home after living in Asia for 10 years. We watched in horror as they tossed our suitcases from the dock into the plane! We had two suitcases filled with bedding and sleeping bags (the best padding we could come up with) with our clocks in the middle, and the check-in people covered the suitcases with "fragile" tape. One of the clocks still works, the other (our favorite) works again after a clock repair person went over it, but the Westminster chimes are so tiny and uneven that we no longer use it. Sigh. I'd be very reluctant to have it in the luggage area where you had no control over it's treatment.
You are going to have to call up the airlines to find out. Maybe if you have no needles, scissors, or.....? (packing all that in your shipped through bags) they might let you take it on. Remember that it will have to go up high and that can be hard with a heavy sewing machine (unless they let you put it in a 1st class coat closet on the floor... but I'm not sure Southwest has first class).
I'd go with a plan B.... rent or borrow a sewing machine while you on wherever you are going.