Dont worry it will be fine...you can gate check it. I took my Stokke Xplory all the time and its WAY bigger than yours. Keep in mind that you wont be picking it up till you get the rest of your luggage though.
Not a big deal on the St Lucia side, but will be more annoying when you come home and have to go all the way through immigration and arrivals before you can get the pram back. Bring a sling or expect to carry a grumpy baby and all your bags at the same time.
Nowadays I use a cheapo Tesco Direct pushchair for airports, and I check the Stokke in the airline approved travel bag at check in so I dont have to deal with the hassle. The cheapo is small enough to fit in the wardrobe on plane. Although if its a full flight with lots of families and you are flying in Y class you might still get it taken away to the hold.
http://direct.tesco.com/q/R.209-1145.aspx