However grotesque the "more better" construction may be, I think this sentence is grammatically correct.
When diagrammed, more does not modify better — it modifies the object, things —, so perhaps you have found a case where "more" + "better" can be used correctly?
"I'm not quite well enough ready yet" is perfectly acceptable in speech, and a stinker on the page because its middle too sticky to parse without cues of intonation and speed. Correctness is a slippery idea in natural languages, be careful with it!
That said, in brief, it is a declarative sentence with a predicate comprising an adjectival complement. At the core of the complement is the adjective ready. This is successively modified by the adverbial phrase of degree well enough (itself the adverb of degree well modified by the adverb enough meaning sufficiency) and then the adverb quite. The resulting adjectival phrase is then modified again by negating it, leaving another adjectival phrase quite well enough ready, which is then modified again by yet (again leaving an adjectival phrase), which restricts the phrase to mean at the present time not well enough ready yet.
[I] ['m [[not [quite [[well enough] ready]]] yet]]
If you are interested in the process or a detailed explanation:
The overall structure is clearly a declarative sentence taking the form of a subject "I" followed by a predicate "am not quite well enough ready yet". The subject is simple.
The predicate takes the form of a present tense form of "to be" followed by an adjectival complement "not quite well enough ready yet". Such sentences are common and used to declare a state. Clearly in the broadest sense, we have a sentence analogous to "I am green" or "I am happy", where I have used a simple adjective to stand for the adjectival phrase.
The adjectival phrase presents us with a problem in that it can be attacked from both sides "not + adj.phr + yet" is a common construction which "negates for the time being" an inner adjectival phrase: "I am not green yet". The "yet" cannot work with a positive adjectival phrase (*"I am green yet") so if you really need binary division, I would suggest that the phrase is first decomposed into a negative adjectival phrase and the adverb "yet". It is clearly an adverb as you could substitute adverbs such as "today" or "willingly" without changing the structure) ("I am not green today", "I am not sad voluntarily").
As I said, that negative adjectival phrase "not quite well enough ready" is decomposed into an adjectival phrase prepended by a negation. So now we have an even smaller adjectival phrase. "quite well enough ready". We know we still have an adjectival phrase because we can still substitute an adjective, "I am not green yet".
This inner adjectival phrase can be decomposed into an adverb "quite" modifying an even smaller adjectival phrase "well enough ready". We can again test by substituting an adjective "I am not quite happy yet".
"well enough ready" as an adjective phrase decomposes into "ready", an actual adjective (at last!) and "well enough" which functions as an adverbial phrase. We know it is functioning as an adverbial phrase because we can substitute an adverb without loss of structure. "I am not quite sufficiently ready yet". (adverbs commonly modify adjectives, eg. "exceedingly beautiful").
"well enough" is clearly an adverb of degree. It can be analysed (if you must) into the main adverb "well" and "enough" which is an adverb which modifies "well" to weaken it to mean only "to sufficient practical degree". We can see that it's this way round because "well" stands by itself "I am not quite well accustomed yet".
Best Answer
First off, it's indeed with, not to. You lodge, or file, or register, or raise a complaint with someone. However, you can also submit, bring, take, or send it to someone.
Here are some stats from the Corpus of Contemporary American English just to drive the point home:
Why? There is really no why, especially not when it comes to prepositions. One is idiomatic, the other one isn't. You learn to use with from your mom because that's what she uses. And she in turn learned it from her mother. That is all there is to it.
This is not really too different from how you use "a car" to mean "a car". It might just as well have been "uma João". But you say "a car". And prepositions in particular are more idiomatic than anything else. They do not have to follow some logic, and indeed often don't. In every language, not just English.
And one last thing. You say "the definition of 'with' is 'accompanied by (another person or thing)'". That is not true. That is one definition of 'with'. A good dictionary will have a dozen more, including one that covers "lodge/file a complaint with". Usage follows definition, but definition also follows usage.
Note how if you say "I'm married with three kids", you don't label yourself a polygamous pedophile, either. And as I commented on that other question, in German, which is closely related to English, one is actually married with someone and it's "married to" that would be ungrammatical. In Russian, you'd be married on your wife, while she is married behind you. That goes to show that there is no one true universal meaning to any given preposition. Current usage = coincidence + tradition.