Okay.
The "U2868 Intel Display Audio" audio device is the Intel HDMI Audio Driver, as discussed above.
If you want to play Hi-res music, you have to use the Realtek driver (output device) and connect the Onyx Studio using the 3.5 mm jack. That should give you Hi-res playback. You should use WASAPI Exclusive mode for best audio, but you could certainly try ASIO. However, with the Onyx Studio I'm not sure you will hear a difference.
When you select the Realtek drivers it sounds like the Toshiba checks if anything is connected to the 3.5 mm jack, and if not, plays to the laptop speakers, since they are the only audio device available.
As soon as you plug the Onyx Studio, or any external speaker, into the 3.5 mm jack the Toshiba plays to whatever is connected. That would be controlled by the Realtek drivers.
Bluetooth only supports up to 48 kHz. If you play via Bluetooth, you have to allow MC to resample to 48 kHz or less. Bluetooth actually only supports 16, 32, 44.1, and 48 kHz I believe, so it makes sense to leave the lower sample rates as they are, and down-mix anything over 48 kHz down to 48 kHz. You would set that up in the MC DSP "Output Format" dialogue.
Bluetooth supports multiple audio formats, known as Profiles. It sounds like your "HARMON (HK Onyx Studio 2 Stereo) [WASAPI]" output device is an A2DP Profile, which simply means it uses better quality audio. Google that (A2DP) if curious. The "Headset Earphone (HK Onyx Studio 2 Hands-Free AG Audio) [WASAPI]" output device is, as it says, a Hands Free Profile, which is a low quality usually mono format. I'm surprised that MC thought it could use WASAPI. I'm not surprised it failed to play with an error. Don't use that output device. If you want to use Bluetooth to the Onyx Studio, only use the ""HARMON (HK Onyx Studio 2 Stereo) [WASAPI]" output device".
So, back to output to the Onyx Studio via Bluetooth is skipping.
I suspect that in the past you haven't made any settings changes to allow MC to resample music to 48 kHz for the Onyx Studio. If the audio was playing correctly via Bluetooth without skips, then I have to assume that the Audio Device you had selected in MC was the "Default Audio Device [Direct Sound]". Then in Windows, the default audio device was set to either "Speakers - Realtek High Definition Audio" or the Bluetooth audio driver, named something like "HARMON (HK Onyx Studio 2 Stereo)".
With either of those two selected, Windows was using Direct Sound, which means that it would resample audio to match the output device. i.e. Down to 48 kHz for Bluetooth. So Bluetooth would work correctly. Windows and/or the audio drivers would be switching output based on whether an external speaker was connected or not, and whether Bluetooth was turned on and the Onyx Studio was turned on and connected.
Basically, Windows was working in the background to get your sound to work, without telling you what it was doing.
I'm brand new to Foobar2000 v1.4.6 and don't know where to look to answer your question. In Preferences - Output the device is Primary Sound Driver, and the Onyx works via BT. If I select HARMON it also works (via BT).
This pretty much confirms my thoughts here. (You posted that while I was typing. Good timing.)
So, what do you do to get Bluetooth audio working again?
Set MC to play the "Default Audio Device [Direct Sound]" Audio Device. You should not need to make any other settings changes, such as "Output Format" mentioned above.
See if that fixes the skipping problem.