I can see two ways to parse this phrase. Neither parsing considers the word "more" to be an adverb. Either parsing might smell a little fishy to you.
The phrase starts with "with". This preposition has an object. One parsing considers "decade" to be that object, and the other, "experience".
Let's start with "experience". That noun can be modified by an adjective, and "more" is a suitable adjective. That gives us the coherent phrase "with more experience". In turn, the adjective itself can be modified by an adverb. This yields phrases such as "with some more experience", "with much more experience", "with considerably more experience", "with slightly more experience", "with enough more experience", and so on.
Here's the fishy part of this parsing: The word "decade" is normally a noun, but in this parsing the phrase "a decade" is used like an adverb and modifies "more". It tells us how much more experience.
If calling "decade" an adverb is too fishy for your sensibilities, then we can simply call it a noun and consider it to be the object of the preposition. The phrase "with a decade" is a sensible and coherent phrase. The phrase "more experience" is also a good phrase. "More experience", however, is a noun phrase.
Here's the fishy part of this parsing: The noun phrase "more experience" modifies the noun "decade". "More experience" tells us the kind or nature of the decade. Normally, we would use a preposition to establish that relationship. The phrase "with a decade of more experience" is a sensible and coherent phrase which expresses the intended meaning. This parsing suggests an elided preposition or a zero preposition: "with a decade [of] more experience".
I don't see a way to parse the phrase in question without using something that smells fishy.
Your proposed "with a decade-long experience" phrase is also coherent and sensible, but it means something different than the original phrase. The authors' opinion of C remains the same even after an additional decade of experience -- a greater amount of experience than they had when they wrote the first edition.