The other day, I heard someone saying, "I'd very much say the opposite."
Is that the alternative way to say "I'd rather say" or is it just one of those non-native concoctions?
I'd very much [...] is the same as saying I'd be inclined to [...] since inclination involves holding something with greater respect.
I'd rather [...] implies a trade-off between two options — the inclination to do something and the declination to do something else because of this.
So these two phrases do not necessarily mean the same thing. Though, loosely, they could be similar to most people in a casual conversation.
"Would" is a verb.
"I'd" is, as you know, a contraction of "I would." Though its use as a standalone is unusual in modern American English (Wiktionary marks it as archaic), phrases like your first example are still a commonality in British English, by my understanding.
In the first sentence, "very much" is an adverb of degree, explaining how much the speaker wishes (or "might wish") to say the opposite.
To answer your question: no, this is not a non-native construction; it is grammatical. The use of "I'd rather say X" as opposed to "I would say X" implies a degree of preference. The first has a connotation that, given the opportunity, you will say X, whereas the second implies that you might be convinced to say otherwise (but you would need convincing). The addition of "very much" further emphasizes the speaker's conviction.