As I mentioned yesterday, I wanted to investigate what the white spheres were in the reflections of my bubbles. While I initially believed them to be other bubbles in the scene, it turns out it was actually the particles themselves. Luckily that was one of the first things that I checked.
I had already turned off the particle's Primary Visibility, hence why I didn't initially suspect they were the culprit. I then unticked everything (Diffuse/Specular Reflection, Diffuse/Specular Transmission, etc.) for the particles themselves and that seems to have fixed the problem. I rendered out a quick test just to be sure that was what was causing the reflections, and I'm happy with the results. Once again this was more straightforward than I had planned, but I'm glad it was a simple fix