Stream Qobuz on Oppo BDP105

You could take the elcheapo route for now.
Google Chromecast Audio mini streamer ( still around $50 on eBay).
This in conjunction with bubbleupnp on an Android device will get you streaming Qobuz.

Or if your rig can use Bluetooth you could just install Qobuz app on your phone or tablet and stream direct from that to your bt enabled rig.
the Oppo is not going to work for Qobuz unfortunately.

Does Chromecast support 24-bit/192kHz streaming? I love my Oppo 105 with the Modwright tube modifications, so I'd rather keep it. I don't need a streamer that has its own DAC or storage, but I guess I could buy a Bluesound Node and bypass its internal DAC.

The Chromecast only supports up to 24/96.
The Node will do 24/192 on coax output I am sure but not so sure on the toslink?
What digital inputs does the 105 have or would you be needing to bring signal in via stereo RCA analog inputs?

-USB Audio: up to 2ch/192kHz PCM, 2ch DSD 2.8224MHz/5.6448MHz
-Coaxial/Optical Audio: Dolby Digital, DTS, AAC, up to 2ch/96kHz PCM
-LAN/ethernet

I would obviously like to stream highest quality Qobuz.

I wouldn't get too hung up on all the high-res stuff.  How it was recorded and mastered matters a lot more than the bit rate. 

The DAC in the Oppo is decent, but it's probably not going to resolve the finest details and a lot depends on the rest of your system. 

The higher bit-rate recordings may sound a little different, but not necessarily better.  I've heard a lot of FLAC level recordings on Qobuz that sounded fantastic and some hi-res offerings that were lackluster.