A chicken baxter is performed by aiming the chicken and cancelling the aim at the same frame as you enter a goal ring. However a certain setup is needed to make it work as of how the sprites work.
The sprite ID of the chicken has to be higher than the goalring.
The video shows how sprite IDs are ordered
Running this script in snes9x can help finding a setup and understanding sprite IDs
Note that any setup as of the moment is purely pulled out of my ass but whatever.
When entering the last area (going through the door), have the chicken you want to baxter NOT as your first egg but rather 2nd-6th. (So far mostly two eggs in inventory been tried).
At the beginning of the first area, dispose of your first egg (usually by doing the gatehack).
When you ride the platform shown in the image to the right, grab those three red coins
And at the end (image below), you want to grab the red coin to the fartherest left though you can grab everything there.
After that, just aim the chicken run into the goal and cancel (by pressing Y) it just before you hit it.
The frame the press Y is shown in image below:
Less stupid Setup
Another way to go that might be easier and simpler is to enter last area with one easily disposable egg (like a green egg, chickens can be hard to get rid off) and your chicken to baxter.
Recommended would be to have 1 regular egg and at least two others where one is a chicken. When you first enter the area, swap the first egg out and use the other to gatehack (or don't). When you get to the end (you've spawned the last flower and red coins) shoot your regular egg (that was your first egg) and then spawn the goal ring and do the chicken baxter.
tl;dr dispose of your first egg just before spawning goal ring. EZ evry time