I will be travelling on southwest airlines with a bass clarinet. What is the best way to get the instrument to my destination safely?
the dimensions of the instrument are bigger than the carry on sizing requirements

The safest way would be to carry it on with you. Check the dimensions of the case and the requirements by the airline. I would press them to allow you to carry it on, though. Those things ain’t cheap! And you don’t want baggage handlers throwing it around.