In singing, it's conventional to break syllables differently from the pattern used in writing, even if the score has it printed according to the 'standard' way. For this reason, dictionaries aren't always helpful: they define syllables according to the conventions for written language.

You probably make adjustments like these all the time without even noticing it: we sing the vowel for as much time as the note will allow us, and put the next consonant onto the next syllable. She's OK.

The 'cleanest' rendition would be "spee-ree-tus" which affords only one (rolled or flipped) "r."

Easier to preserve the rhythm and avoid the Chrysler-starter-motor RRRRRRRRRRRR

There are two other issues in the pronunciation of spiritus. 1) Even if you persuade your singers to sing a single flipped r or even a soft d, they will sometimes make a diphthong out of the first vowell: spee-uh-ree-tus. 2) The accentuation of the dative spiritui is correctly spi-RI-tu-i, but very often it is spi-ri-TU-i, in spite of the correct accentuation presented in such chant books as the Liber usualis. Some composers of the sixteenth century clearly set the third syllable as accented, so it has been a problem for some time.

Regarding Prof. Mahrt's observation of unintentionally forming a diphthong as in [spiəritus]...in such a situation it may be possible to correct by pointing out the phenomenon to the singers, then rehearse [i:ri:ri:ri:ri:ri:] with the explicit instruction to keep the shape of the mouth and lips completely consistent, and only move the tongue in order to flip the [r]. Then widen the exercise to include the whole word "spiritus," then the whole passage, etc. If you have the time for that sort of thing.

Another exercise is to rehearse extensively on vowel sounds only before adding consonants. Then add the consonants gently.