tl;dr – From a functional perspective, thankyouverymuch is an evaluative adjunct (a type of stance adjunct) according to Downing & Locke because it is “attitudinal, reflecting the subjective or objective attitude of the speaker towards the content and sometimes also towards the addressee” (2nd ed.; pp. 73-74, 234). According to the Longman Student Grammar of Spoken and Written English, thankyouverymuch is an attitude stance adverbial, which “convey an evaluation, or assessment of expectations” of the speaker’s attitude toward the proposition (p. 384).
From a discourse perspective, Blommaert (2005) would probably see thankyouverymuch as a performative element and a way for speakers to mark an orientation to what they have just said. But I’m not great at discourse analysis, so please tell me more in the comments.
Finally, syntactically, thankyouverymuch is a finite clause.
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