Good question.
You just talk to the airline. I had a friend always booked a seat for his guitar so that he did not have to check it.
All airlines have some policy regarding such "plus-sized" passengers.
Options are trying to shove 300 lbs into a 17" seat ... not pleasant on a full flight, for the pax or their seatmates.
So the idea of booking two seats together, as this lady did, is the proper way to fly on todays' crowded flights. From Southwest:
Customer of size and extra seat policy
In the "Passengers" field, indicate the total number of seats needed by selecting that number of adults. (For example, if one Customer of size requires two seats and is traveling with one other adult in their party, select "3 Adults.")
Proceed through the "Select Flight" and "Itinerary and Pricing" pages.
Complete the "Who's Flying?" name fields for a Customer of size as follows:
Without a middle name: A Passenger named Tom Smith would designate Passenger One as "Tom Smith," and Passenger Two as "Tom XS Smith" (first name Tom, middle name XS, and last name Smith).
With a middle name: A Passenger named Tom James Smith would designate Passenger One as "Tom James Smith," and Passenger Two as "Tom James XS Smith" (first name Tom, middle name James XS, and last name Smith).