Welcome to the forum and a huge congratulations to you! Welcome to the wonderful world of horse-ownership, it's great isn't it? :) BTW we love pics here!
As for the bit, it depends...
When you ride in the Pelham are the reins on the upper (snaffle) ring or the lower (leverage) ring?
neck rein with loose reins in any bit in the world, is your horse responsive to neck reining with the pelham?
That being said I just hate broken mouth pieces on bits for neck reining. I'm gonna just copy and paste what I write in almost every forum asking for a good western bit and my opinion in why you shouldn't neck rein in a bit with a broken mouthpiece:
"With a Curb bit that's mullened out or has a port and is not broken here's how it works. You neck rein left, your right rein shortens, the right rein pulls on the right shank, which twists the bit and pushes the right side of the horse's face and applies pressure on the left bar. This is very clearly pushing and pulling the horse's face. Eventually horses learn the pre-cue of the rein touching their neck so your reins can be long and barely touch their mouth at all because the horse knows and understands, at this point you can use whatever bit you'd like.
If the mouth piece is broken and has shanks (like a tom thumb) When you neck rein left, you right rein shortens, it both pushes and pulls on the right side of the face - not touching the left side of the mouth at all. While it has much stronger brakes - causing the nutcracker action with the leverage of a curb is a very strong bit. But the steering communication is NOT clear for neck reining. Of course if the horse neck reins solidly and your reins are real soft it doesn't matter, but it's still a strong bit.
Just to be clear I was comparing those two styles of bits apples to apples, assuming the same amount of pressure and the same shanks and purchase - the only variable being the mouthpiece. I am well aware that bits are only as soft or as hard as the hands holding them, but I'm just comparing them apples-apples. :)
This is my favorite bit for neck reining :)
There's no need to find a bit that's similar to a pelham, the more important aspect is to find a bit the horse is comfortable and responsive in. Do you have a trainer or someone else who's knowledgeable helping you out? I'm sure they'll be able to help too, knowing your particular horse better than I or anyone else on the internet could.