Syllables are largely unnecessary to describe the ...i_e spelling pattern
The textbook is correct in stating that the spelling pattern found in fire, wire, tire can be equated to the spelling pattern found in bite, kite, site: in both sets of words, we can view the "...i_e" pattern (with a single consonant letter in between) as a spelling of the diphthong /aɪ/ (pedagogically called the "long i" sound, due to its spelling typically involving the letter I).
Bringing syllables into it may not be wise. There is much disagreement about both the theoretical and actual nature of syllables; while in most cases people agree about the number of syllables in a word, there are some cases where they don't (disagreement may even be found between speakers of "the same" dialect, not just between speakers of different dialects).
I'd guess describing this as a spelling pattern that applies to "one-syllable words" is intended to exclude one of the following categories of words:
words like image, where there is another vowel letter in between the I and E, which never follow the ...i_e pronunciation rule (except by coincidence)
words like infinite and feminine, where the final syllable is unstressed, which sometimes do and sometimes don't follow the rule
words like mineral or literature where there is an "...i_e..." sequence in the middle of a word, rather than at the end: these generally don't follow the rule (unless the word happens to be a compound)
But given that there are in fact words of more than one syllable that use the "...i_e" = /aɪ/ pattern (such as inspire, divide, organize) I think it may not be useful to instruct students that it especially applies to one-syllable words. Rather, it seems more to the point to state that it is a spelling pattern for the ends of words, when a word ends in the letter I + a single consonant letter + a silent E letter. (The fairly small number of exceptions to the single-consonant-letter pattern, such as title, idle, trifle, blithe, tithe, are not truly irregular, but may not be worth accounting for in a simple statement of the rule, since the different consonant groups that can appear before silent E follow different patterns (as with bridge, prince, wince, singe) and there are some words with truly exceptional "short i" such as triple).
Speakers who use two syllables in fire add the second syllable automatically
For speakers who pronounce fire with two syllables, the second syllable is either a syllabic /r/ sound or a schwa /ə/ followed by the consonant /r/ (or just a schwa /ə/ in "non-rhotic" accents like typical southern British accents).
The use of this syllabic pronunciation instead of a non-syllabic /r/ sound is pretty much predictable based on the sounds that are in the word: generally, speakers who use /aɪər/ in a word like fire will also use it in other words like hire, pyre, dire, and there won't be any particular words where they do use single-syllable /aɪr/ as a distinct pronunciation.
This means that (at least for speakers who have an /r/ sound in these words) we can view the schwa as something that's automatically inserted to break up the sequence /aɪr/. A linguistic term for the insertion of a vowel is "epenthesis", so we can call it an "epenthetic vowel".
Epenthetic vowels like this can be potentially found in a number of contexts in English before either /r/ or /l/: after a vowel that is a phonetic diphthong (such as in hour, wild, owl, oil) or in the consonant cluster /rl/ (as in world, Carl, snarl).
These epenthetic vowels are of linguistic interest, but since it doesn't make a difference to the meaning of a word whether they are included or omitted, you'll only need to explain them if your child is actually puzzled by the fact that they often have no explicit representation in the spelling of words. But if your child is confused by fire being included in the same set as the other words, it seems fine to set aside -ire words as a separate pattern. I wouldn't say you need to insist against the evidence of your ears that fire has one syllable; rather, the point is that it can be sounded out as an /f/ sound followed by the "long i" sound followed by the /r/ sound, and the /r/ sound may create a second syllable after the "long i" sound.
Prior questions on this site about this topic: