G’day Lachlan,
There were no tricks, trial and error, guesswork or great skill required in fitting the “birdsmouth” to the other length of pipe. All I had to do was set the “swept radius” of the cutter to suit the dia/radius of the pipe and wind the handles. The lathe did all the rest.

No patern, template etc. was needed.
The pipe to be “mouthed” was simply set up as shown in the photos. The centerline of the pipe was set to the exact height of the lathe centerline. This was done by using the two steel plate "fixtures" as shown, no problems. Now that I have these fixtures, I can "mouth" any pipe that is less than 2.374" (60.3mm) in dia. by simply making two spacer rings to fit.
I am fitting a nylock nut into the joiner and a recess has already been cut into it for the nut to be set in. If the nut is too wide (thick) I will thin down a standard nut and keep it there with a drop of loc-tite. The cap head screw will be accessible and can be removed with little trouble. I have also up-sized the screw to M16-2. I would have prefered to use a 5/8 dia BSF thread but the BSF series is no longer made. However, the M16-2 is a finer thread than the 5/8-11 UNC.
If you are having difficulty in shaping the “mouths” to be the same shape and form, you could try using a hole cutter if one is suitable for the size of pipe/tube you are using. The set up needs to be very secure and run the hole saw at a slow speed, around 50 feet/min. max. This technique works very well with aluminium tube. The only flaw is the initial cost of the saw. If you could find a cheap way to transport your pipe to me and back, I would cut the “mouths” for you, gratis.

Kody