Ok so my experience comes from catching chickens and clawed ducks as a child, so assuming you're a full grown adult, and this chart, the ratios are the same.
You gotta catch them from surprise, from the back, but it sounds like you're already in the fight if shit goes down. The beak is your issue. The wings are just a distraction. Get the pelican bastard from the neck, as high as possible if you can and try to grab the legs. ChatGPT says they don't really use their legs to fight, but worst case, start swinging it. I bet once you clamp on the beak, it'll be hard for it to open. Like how alligators can chomp down, but have trouble opening. Once it's subdued, it might stay freaked out for a while. You just gotta hold it until it accepts defeat.
Then take it to your mom and she'll take the head and feathers off for dinner.
"shocking" is a strong word