Yes you can play back your DSD downloads on a Bartok. I do mine. My DSD files ( together with all of my PCM rips and downloads) are all stored on a Melco streamer which I basically just use as storage pretty much along the same lines as your Core. I checked the Korg website and it looks like it caters for all common file formats. However the website is a trifle ambiguous. Nevertheless you should be able to copy files from the Korg to other media and also convert the files to other formats if required. There should be lots of flexibility to find you a better solution than being trapped inside your Korg ( so to speak).
I have never yet found any example of an album on Qobuz where I cannot identify -from Qobuz itself- which recording of any available multiple it is. Yes there is a provenance issue at a deeper level e.g. is this a new digital remastering or have they used the production master that was used for an earlier digital issue? However at that level there is no advantage with a hard copy as the supplied data that accompanies it is usually the same as the virtual copy of that is supplied with the download or can be accessed online through Qobuz ( some box set material excepted) as all comes from the record label itself. A download from Qobuz is the same as a download from, say. Presto Classical. There is no intervention from Qobuz either with streaming or with the download store. They just provide what the record label sends them. I don’t know when you last tried it but if before mid 2017 you may have been listening to MP3s or at best 16/44.1 and not <24/192 as is the current case.
I won’t go on and on but I still think that you will be missing out and that all of your objections which may have a technical basis can be overcome.
Anyway good luck and please let us know how you progress.