Preface: I own neither the Prophecy Type-R nor the Atch Attachments kit.
1. From the pictures I have seen of the Type-R internals, I doubt that the slamfire section would fit in easily. The retaliator has specific sections for the locking system – which the Atch kit appears to rest on. The Type R does not have those ridges, as it was never designed to accommodate locks. With putty/adhesives/polycarbonate work you could probably get it to work and sit in there, but would mess up the shell and also that brings me to problem 2.
2. You have to cut down the boltsled a fair bit on a retaliator to accommodate for the forward motion of the slamfire. I have my own doubts about the long-term durability of the Worker blue plastic sled in its stock form at higher spring forces – I would be VERY hesitant to cut it at all. A solution would be to purchase a Roboman/Artifact/metal sled and cut that down, but that's quite expensive.
3. I am of the opinion that slamfire for ~14kg and up is difficult because of the sheer mechanical forces required (I'm 20 and been dealing with homemades for years, slamfire on a 14kg system sounds like hell for longer games).
I doubt that slamfire is going to vastly increase your ROF to a significant degree. ExPT retaliators (as a group) occupy a weird upper niche of 180-220 fps (lower power ones get 150+) which is great for a 200fps cap game, but ultimately being chipped away at by flywheelers (OFP's 180-190 fps Eclipse system for example). TL;DR focus on accuracy and power.
4. Misc: the Worker Prophecy Type-R breech is not compatible with Jet Katana Magazines with the included short-dart breeches. Will only work with Worker/Artifact magazines.
The Worker "long" 25N spring is not the commonly-known [k25]. Easy mistake, I was weirded out too.
Furthermore, the Atch Attachment kit appears to no longer be avaliable on Etsy.
If you decide to go forward with this, best of luck! Excited to see what you can come up with.