I know there's an exception for "melodic minor" (M7 ascending, m7 descending), but does the framework of musical modes support the notion in general? Put another way, are there any modes that have multiple notes for a given degree regardless of the direction a player is going diatonically?
In the way the question is put, no. A mode containing, for example, both E♭ and E natural when ascending would consider those separate scale degrees and would number them accordingly.
Wikipedia provides a useful definition of mode:
In the theory of Western music, a mode is a type of musical scale coupled with a set of characteristic melodic and harmonic behaviors.
The diatonic modes all adhere to the rule that they have exactly one note per letter name, so for those modes, the multiple-notes-per-degree situation can't happen.
An exception (sort of): The "bebop" scale
The "bebop" scale is an eight-note scale containing both the minor and major sevenths above the tonic. The F bebop scale, for example, contains both E♭ and E-natural, and those are generally talked about as flat and natural (or minor and major) seventh degrees, respectively. For the purposes of this question, the bebop scale can be viewed as a "mode" insofar as it has a reasonably well defined melodic usage (though not harmonic), but in practice it's a modified mixolydian scale. Its typical usage is as a melodic path through a dominant seventh chord.
Melodic minor, by the above definition, qualifies as a mode and therefore satisfies the question. It's the one mode/scale having distinct pitches ascending and descending.
It is not, however, generally thought of as a mode. It's more often just considered a variation of the diatonic minor (which is a mode), reflective of common compositional practice (leading tone when moving upward to the tonic; no leading tone when moving downward away from the tonic).
Octatonic (diminished) "modes"
The octatonic scale could be considered as having to "modes": the half-whole mode and the whole-half mode. The harmonic behavior of the octatonic scale isn't particularly defined (certainly not in the way the diatonic modes are defined), but they do vary from each other in their melodic characteristics. At that level, we might consider them "modes", and there are reasons they might be written using enharmonically equivalent notations depending on context (such as sharps when ascending and flats when descending). However, the pitches remain the same regardless; only their notation changes.
The whole-tone scales do have a more defined (or at least, expected) usage, so could be "modes" in the sense given above. The whole-tone scales also might be written in enharmonically various ways as a matter of convenience. But they also would be equivalent in pitch, varying only in the notation used for those pitches.
though this is matter of interpretation.
For example, in jazz terminology melodic minor denotes the following scale: 1 2 b3 4 5 6 7, regardless of the direction.
Seventh mode of melodic minor would be then: 1 b2 b3 b4 b5 b6 b7, however the most often it is thought of as: 1 b2 #2 3 b5 b6 b7. So rather than having minor third and diminished fourth, flat and augmented second (or ninth) coexist in the scale and the third is minor. The scale is also called superlocrian or altered and is a common choice to play over dominant chords with corresponding alterations.
This is a subject of interpretation, because it's up to you how you notate the notes, but using the scale in a way that the audience hears diminished fourth rather than major third is a nontrivial task.
If you explore modes of less commonly used scales, you will find more examples.