I found several examples of tone being an "integral part of" a morpheme or a syllable, but not of a word.

Mandarin is a tone language in which **each stressed syllable has a significant contrastive pitch which is an integral part of the syllable**. ...
www.infoserveusa.com/training/chinese_mandarin.htm

When added to a word that ends in a Lo tone, as in the examples in (1), the tone of ..... and that ** the Lo tone is or was an integral part of the morpheme**. ...
links.jstor.org/sici?sici=1477-9366(1992)1%3C67%3ATPRMIH%3E2.0.CO%3B2-%23

But in Hoyloy they [syllabic tones] are consecutive elements each forming **an integral part of each syllable in a word. Tones used in Hoyloy are thus named as syllabic tones.** ...
hoyloy.twintl.net/trans-1.htm

Or maybe "integrated into the verbs" - which is not as strong a statement:

This morpheme is the only means of expressing the reciprocal meaning in the ...... from **a pronominal proclitic integrated into the verb morphology**. ...
www.stanford.edu/~emaslova/Publications/Reciprocal.pdf

In any case, I like using the idea of "integrated/integral".