We use the solfege associated with the clef, so for Angelis suis we would use SOL as the starting pitch. Similarly, if we are using solfege with polyphony, we would use the key signature as the DO (so a key of a minor and C major would both have a DO of C). We do a fair amount of chant alternated against polyphony, so over time the expectation is for choir members to use solfege to connect through the transitions.

There may be an occasion where a transposition occurs between Gradual and Alleluia / Tract... where we read the DO clef as a FA clef or vice versa. An example might be if our ending pitch of the Gradual (DO clef) is SOL and we are transitioning to an Alleluia (in mode 2 FA clef, say) with a starting pitch of DO... in which case the DO of the new key might be the SOL of the old key. In that case, we may demonstrate the correspondence of the old modal scale to the new modal scale... but typically that is only long enough to demonstrate the transposition.

We would simply call that flatted-ti "te."

Is this modified if there is actually a B-flat key signature, e.g. the solemn tone Alma Redemptoris and Regina caeli? Does B-flat become fa and F do?

I think not. Alma Redemptoris (Mode V) begins on fa, the first few notes being fa la te do do re fa fa. I think you are trying to force the modern concept of key (signature) into chant notation here.

I understand, but in movable do with modern notation, a key signature of one flat means that B-flat is called fa instead of te.

I don’t follow this logic. (Isn’t the 7th scale degree the only one that is ever altered in chant? “Fa” might become home base in a chant depending on the mode, but it doesn’t stop being “fa” and “te” doesn’t stop being “te” either. It’s tempting to want to pretend that it is in “fa major” perhaps, but then you are confusing the two systems. [and it’s not actually Fa major.])

Movable do is my system of choice for all tonal music because it more accurately represents the function of the tonal system; however, for modal music such as chant I find the "fixed do" system far more intuitive.

"Fixed do" in Gregorian chant - do always refers to the note indicated by the C clef/fa always refers to the note indicated by the F clef, regardless of the mode of the chant or key signature.
"Movable do" in Gregorian chant - do refers to the finalis of whatever mode is currently in use.

I use whatever is easiest, and usually both systems within the same chant. Often times a different modality is established for short sections, so I change "do" to whatever scale degree makes singing the intervals easier.

For instance, in Mass I, I would start on "mi", and then at the "Pleni sunt" I would do "do-sol-le-sol-fa-me" and then at "-ria tua" I would sing "re-re-mi-la-sol-sol."

I would do the same for modern keys and switch up solfege if it makes it easier to sing phases with alternate tonality, secondary dominance, etc.

I don't think "movable do" is a term that quite encapsulates solfege for chant, since tonal harmony has only the named key or its relative minor, whereas in chant there are usually more than these 2 tonal scales that roughly approximate the mode. For instance, in the Lydian mode, usually the easiest thing for me is to call scale degree 5 "do", since Lydian approximates G major.

Yes, in a fixed do system "do" refers to the pitch of C, no matter the key; whereas in a moveable do system "do" refers to the tonic pitch.

Restating the question as clearly as possible:
With a key signature of one flat, if we call F fa, B-flat te, and C do, is that an example of fixed do or movable do?

This misses the forest for the trees, though. The modes denote scales and their inherent pitch relationships. For me, the most valuable tool of solfege is the aural recall of intervallic relationships between scale degrees. The problem is that the majority of my training and the vast preponderance of my aural exposure has been to tonal harmony and mode I, the tonal scale. Thus it is far easier for me, and I would expect for many, to relate the intervallic relationships between the other modes to mode I, with tonic as "do."

When there is clear aural recall of these intervallic relationships, it becomes easier to apply a mode 1 mental frame of reference to short sections of music, and to call any pitch class of convenience "do" where its tonicity has been established

I try to chant Vespers (in the middle of the week) and keep a fixed do when I do that, across all the chants of the particular hour.