You can do pretty well with SWISS emulator with ActionReplay. However, the downsides are that the AR doesn't read SDHC cards, so you are forced to run SWISS on a low capacity SD card, then use another memory card reader/port for a larger capacity SDHC card. It's definitely not ideal. However, the way the GBC works, you COULD produce an updated version of the AR software with SWISS, or an offshoot, that could read larger capacity SDHC/SDXC cards from a single memory card slot. Then sell the disc and the reader (basically the WiiKey). I feel like SWISS compatibility is pretty good, to be honest, though it's not perfect.
There are some homebrew options to replace the DVD drive, but it requires modifications, and isn't that feasible. Unfortunately I think most of the WODE hardware (like the Wasp) are no longer being produced, at least that was the case as recently as 2017.
Now, as for a direct device that did not require drive replacement, the Cube actually has several existing ports. The Parallel Port, used by the Gameboy Player, Serial Port 1, used by the Ethernet Adapter, and Serial Port 2, which was used by developers for USB access. SP2 was removed in later systems, similar to how the Digital Out was. Anywho, the serial ports use EXI protocol same as the memory slots, so technically you could build something that took an SD card same as the AR or WiiKey does. Serial Port 1 makes more sense, as really nothing uses the BBA anymore (beyond some PSO), and there's more room to work with to fit an SD adapter of some kind. The downside though is that both the serial ports and controller ports don't have a great bus speed. I've read that you're probably talking about being 4x-8x slower than the DVD drive. Again, the AR option still allows for decent compatibility though.
The other option is to use the Parallel Port instead of the Gameboy Player. There are some advantages there. For one, that port is truly "hi-speed" and you would have more than enough bandwidth for any game. You could also incorporate the existing GBP case design (injection or 3D printed) to fit such a device. The downside is you'd "lose the GBP," although I guess an emulator or a possible hardware passthrough to "stack" one device on the other could be done, but at more cost.
