Studio One is not able to store mix settings used in a Series III mixer. Only Capture has this feature and it needs to be saved separately from the recording. Once the "recording scene" is saved via Capture, it can also be loaded in Studio One, but - as you write - the processing settings are gone. Only "Fat Channel" is already inserted to all tracks, but it is "empty". The "Fat Channel" is actually the only processing the mixer and Studio One share. In Studio One the Fat Channel is a plug-in and a "host", because there are now the plug-ins of the "Fat Channel Collection", which you can use in both worlds.
A typical and more traditional approach would be to record tracks in Capture and load the raw files in Studio One and do all processing and Mastering there. The worlds are clearly separated and you know what is happening where. A different approach would be to record tracks directly in Studio One as raw signals and then change the mixer to DAW Mode for mixing the recording. This would allow you to use the mixer's processing and effects during the mixdown and simply record the mix in a stereo track.
It would be really helpful if Studio One would be able to receive channel/track names as Capture can do it, but on the other hands it's not more than naming channels and tracks individually. That would be 24+2 in your case, so it should not be too much work...