I have the Lunt60, but unfortunately different telescope, focuser etc, so I don’t have direct experience.
Since the 174mm mini is 8.5 mm distance to the sensor and the 585MC is 6.5, so all else being equal you would just need to focus out 2 mm. However, I believe the mini is a female attachment, the 585 is male. So how are you attaching the cameras to the blocking diagonal, you must need a different adapter for the two?
Are you having trouble focusing in far enough, or focusing out far enough? I recall seeing somewhere that the Lunt50THa can have trouble with some cameras with insufficient backfocus, so they could not focus in far enough. As I recall they screwed the camera directly into the diagonal, after removing the adapter on the diagonal.
Another solution is to try a short barlow in the diagonal, that may extend your backfocus if you can get the barlow right into the diagonal.
But I have a question, why are you using a colour camera. You ideally want to use a mono camera, you loose too much resolution and too much signal—signal is really important for solar, with the filter there is not much light, and you want your exposures to be as short as possible. And you are trying to focus on a red light image, as blue and green will be totally black. This means you are only using ¼ of the pixels, the Bayer process will try to fill in the other pixels, but you are losing a lot of resolution. So it may not be very sharp because you are skipping pixels—your 350 mm is giving you a resolution of only 5”. And red is not a great colour for the eye to focus on either. Could this be the issue?
Rick