Well, that's actually a complicated question. It depends on how you analyze things. Nominalized clauses (e.g., with "that") act like nouns. So the distinction between CP and NP can seem to fade away.

Another way to address this is to question what type of clause that is. Are you considering it a relative clause modifying 'fact'? In that case, your second approach makes sense.

So the first approach seems like it might work, but probably not quite like that. The problem is that even if you wanted to consider that whole structure a CP*, it's not clear what position within it NP would take. It's not the subject or topic within that clause (right?) so it doesn't seem like it would fit with that structure.

In summary, there are two ways to address this:
1. Consider structural and semantic analyses that clarify the best representation.
2. Focus just on constituency, such as matching up heads and phrases to see what should go where. This is probably where your instinct to switch to option 2 came in.

By the way, looking at heads and phrases for constituency can actually be easier if you draw out the full trees, so that's one reason to do it even if you're confident about the structure within those phrases above. (I realize it's time-consuming to write out the trees here, so I'm not objecting to that, just mentioning it for your own notes as you sketch these things out. Even just indicating the head by underlining it or something could help you to think about these issues.)

[*Indeed, a CP can act as the subject, because "the fact" is optional! So this is on alternative hypothesis you could consider.
Actually, one further possibility would be to extend this by analogy to assume that even without "the fact" there is some noun head there converting the CP to an NP, so you'd have to imagine a silent "N" element that takes the same position as "the fact" in your second approach, and that might be a reasonable way to go. Of course some might say that "that" actually does that itself, although it's then debatable whether the resulting structure is a CP or NP. These details can get complicated.]

Does "one cat" function more like "one" or "cat"?

Edit: I should also add that this is a slightly more complicated question. It really depends on your theoretical analysis. If you have a Determiner Phrase (DP), you could call the numeral a determiner. You could also add a layer of structure below DP that specifically allows for numerals, and you could call that a NumP, if you wish. But in that sense, it's acting like a quantifier, which is arguably a type of determiner. All of these may be split into different categories based on nuances, depending on the analysis. If you're simply treating the numeral like an adjective, then like an adjective, this would be a modifier phrase within the NP. But if you get into more complex phrase structure, you could say there's a functional 'NumP' projection within the larger DP/NP structure. So, it's complicated, and it depends. As with my (over simplified) original question above, what sort of tests would be relevant for determining these differences (and what assumptions are you making)?

Given that your trees above just have "the fact" as an NP (not a DP), then you would also analyze a numeral as within the NP. If you do introduce DP, then "the fact" would be a DP (with "the" as the D head), and then the numeral would be a part of the DP (or a more nuanced projection within it). But if there's only an NP, then that would contain the numeral, rather than the other way around. Essentially an NP is a primitive DP, and a DP is just a bigger NP, in terms of external function.

Those are great observations/questions.

There's no "right answer" here, just what others have done, and various motivations (such as the data you just introduced).

If this is for a class or other situation where a theory is assigned to you, follow the structure there.
If this is for your own project and you can choose (or change) a theory, then you should read about "DP" as I mentioned above. It's standardly assumed in current Minimalist syntax, and has been for decades. So instead of NP, you'll have DP as your noun arguments in a sentence (subject, object, etc.), and then NP will be the basic structure around nouns including modifiers like adjectives or prepositional phrases, etc. Then above that you'll have additional layer(s) including the D (Determiner = articles) level.* And you can add a NumP or QuantP (or whatever else) as needed. Again, refer to current research for examples and arguments. The easiest way to approach this is of course to just follow the structures assigned by others rather than reinventing the wheel, not that what is already out there is perfect, but it's probably usable.

Your question about complex numbers is very important. This means there is internal structure within the numeral. Therefore, one possibility is to have two distinct "numeral phrases". That is, one like an AdjP where you have the structure of the numeral itself, and then some higher level of structure that hosts the NP. That gets a little complicated though because you might want the Numeral head to also select the NP as complement. In that case, you'd have to put the modifying numeral words somewhere else. There are several possibilities: the "specifier" position might be somewhere to put them, but then they could be complex, and at least semantically it isn't clear to me why you'd want to split up the words in that way as if for some reason "twenty" is more/less important than "two" in "twenty-two".** So another simple option would be to say these are actually compounds (morphology) rather than syntactic constructions, or even if they are syntax, treat them somehow as complex heads. (That gets into some technically complex issues with drawing and theorizing about syntactic trees, but it can be done.)

In short, it's time to do some reading to figure out which theory (or really, theoretical variant) you'd like to follow.

--
[*But note that because articles also agree with the noun in gender, that doesn't necessarily mean your reasoning for treating ordinal numerals like adjectives is correct, although my intuition is that it would make sense.]
[**Actually, agreement patterns in some languages could give us a clue: sometimes a number like "twenty-one" would trigger singular agreement, I believe, although I can't think of an example right away. If not, maybe there is some other way to distinguish between the parts of the number with one as head, but it's hard to say. Another complexity is syntactically complex numbers like in German literally "one and twenty", e.g., Zweiundzwanzig, although note that writing it as a single word might suggest this too is a compound.]