Theoretically speaking, using a T-mount lens to shoot an eclipse should create no such issues related to infinity focus/image quality/etc. Most DSLRs that shoot through a telescope are attached via T-mount adapters and, depending on the quality of the telescope's optics and the technique of the photographer, images taken using such an arrangement can be quite spectacular.

Infinity focus is not an issue because the T-mount specification has a flange focal distance of 55mm. Most DSLRs and even older 35mm film camera systems have FFDs in the 40-48mm range.The adapter that screws onto the 42x0.75mm T-threads needs to be anywhere from 7mm to 15mm thick to fill in the distance between the camera and the lens. Thus no additional optics are needed to achieve infinity focus.

Of course there is no communication between the lens and the camera so you need to be sure to use a camera that can shoot in Manual exposure mode without a lens detected. There are a few entry level cameras from various manufacturers that do not have such a capability.

You'll also need to focus manually, but for astrophotography that's not much of an issue since 1) Most astrophotography is done at infinity focus and 2) Most camera's can't AF on objects in the the night sky other than the moon. Depending on what type of solar filter you use to image the sun, even if you had an AF capable lens it might or might not AF correctly through the filter. In any case careful manual focus, typically using magnified Live View, will usually get better results with astrophotography.

In the real world, though, things are a bit different. Most mirror lenses designed to be primarily a camera lens that have a T-thread at the rear of the lens for use with cameras via a T-mount adapter aren't that good optically. This is particularly the case with the cheapest ones. You're also limited to a single aperture with almost all such lenses, which may reduce your options regarding ISO and shutter time. Note that cheap refractive lenses intended primarily as photographic lenses that have T-mount connectors are even worse optically than their mirrored counterparts.

Higher quality mirror lenses tend to be made with specific lens mount connectors permanently attached to the rear of the lens, rather than T-threads for a T-mount adapter.