Is 'who' here a relative word or an interrogative pronoun?

"Who" is a relative pronoun, not an interrogative pronoun. A relative pronoun is used to connect a clause or phrase to a noun or pronoun. Who, in fact, is one of the most common relative pronouns. An example is - "The driver who ran the stop sign was careless."

References - https://grammar.yourdictionary.com/parts-of-speech/pronouns/relative-pronoun.html

I am not a literacy or grammar expert, but for my own use I have (perhaps from a teacher long ago, who knows) a simple rule that if you would use the word "him" (that is ending with the "m") then you should use "whom". I think Important people will have all sorts of names for the category of word but I just find that intuitively one can decide if it would be "he" or "him" in the sentence if it were unscrambled.

so "who I am" you would re-arrange it and using he/him it would read "I am HIM" not "I am HE" and thus I would suggest, tentatively and standing to be corrected, and all that sort of thing, that most of the question relates to sentences that would require WHOM. of course some of these sentences would appear clumsy so we wouldn't say it at all.
Basically it could be both. Does this help?

"who I am" as a declarative noun clause – object of "of" – would seem to rule out any interrogative function, actual or virtual.

I believe they are both relative words.

A relative word is used to refer to a subject that has already been mentioned. For example, in CaGEL example i and ii, the subject was represented by the relative word "What".

So if you were to replace the relative word "What" in example i and ii, you would end up with the same thing-"A faulty switch caused the trouble."

We can split "That's a big part of who I am." into: "That", which is our first subject, "is a big part of", which are connectors and the like, "who",our relative word, and "I am", our second subject.

Together, "That who I", or "(Subject 1)who(Subject 2)" So, I believe that who is a relative word in this situation.

The confusion over this may be because of, firstly,"That" being a relative word by itself, which relates to a previously mentioned non-human subject, and, secondly, the word "Who" being used to relate to, again, a non-human subject, and lastly, "Who I am", being a phrase.

"When that day comes if you don't like who you are, you're done." seems to be quite obviously relative. "Who you are" is the subject, where who refers to the "you" in question.

My final argument would be that the word "Who" comes in "Who you are" and "Who I am". It is clear these are relating to You and I. Their counterparts, "Who am I" and "Who are you", are the ones that contain the interrogative "Who"s, asking about the identities of You and I.

Hope this explanation helps everybody.

(1) That's a big part of [who I am].

Relative pronoun. This is a statement, not a question."Who I am" is a noun clause that is the object of the preposition "of". (I am who)

(2) When that day comes, if you don't like [who you are], you're done.

Relative pronoun. This is a statement, not a question."Who you are" is a noun clause that is the direct object of the verb "like". (you are who)

Indirect questions.

Your question is an interesting one, and I am not surprised it is still floating around.

CGel can reasonably treat the who used in this type of context as a relative pronoun. It would involve something like understanding an unstated "..is who I am". That is a reasonable way of looking at it. Similarly, you can argue that it operates as an interrogative pronoun. In that case "who I am" is indirect (reported) speech, where we are to understand "..is who I am." Either is defensible.

The first and obvious thing to say is that despite the now almost universal use of this fashionable idiom, dictionaries like Merriam Webster and the Cambridge English dictionaries do not seem to deal with it. I can tell you what it means (as if you didn't know already). Whenever its first use may be, I doubt (from memory) whether it can be found before 1990, and possibly not before 2000.

Understanding the expression has very little to do with grammar. I mean that whoever its first users may have been, they are unlikely to have had a precise grammatical analysis in mind. And whoever they were who caught onto and spread it around the English speaking world were not interested in the grammar of it either. The fact that this saying is grammatically troubling is probably one reason for its catching on and spreading so widely. What it is, I have learned from internet searches, is a meme, and indeed a locus classicus of a meme. (By the way, locus classicus was probably a sort of meme of its own time (which ended during the second half of the 20th century). It was known then (in my younger years) as a Latin tag, widely used by academics barristers and senior civil servants till the 1980s, when the British civil service and other banned them from their publications and communications!)

The expression "it is who I am" is used more than once by Chris Tomlinson, who has a song which uses the meme over and over again:- https://babylonbee.com/news/im-chris-tomlin-its-who-i-am-16x---op-ed-by-chris-tomlin. Every verse consists of one repeated line as does the chorus (refrain)

I'm Chris Tomlinson, it's who I am.

I shall not discuss the literary merits of this particular use of the meme. But one thing is obvious, and that is that there is nothing grammatically troubling about it. The use of the word 'who' may be interrogative or relative. It tells you the identity of the speaker. Once I know your name is Chris Tomlinson, I know part of how to find you again. Though there are many other people of that name, it's a good start. It gives me information about your identity.

Then there is a song by Chris Young called Who I Am with You. The song begins with an account of his making a bad start in life: a rolling stone, and in general not much good.... until he meets his beloved, when, in a rising climax of volume and pitch, he declares that

A better man is who I am with you.

This is different. This is linking his quality as a person to to his identity. But of course, if you ask the lucky girl who her boyfriend is, she is not going to answer: "He's a better man", or "a good man". She certainly is not going to have much luck asking any of her friends whether they know where this better man she met up with on holiday might live. "Do you mean to tell us you didn't find out who he was?" they might ask incredulously. So there is something queer about this locution.

The expression is often used to lay claim to particular personal qualities as their identity, or part of it. Someone can also use it to assert that some pursuit or avocation is so important to them as to be part of their identity: "musician/plumber/safe cracker/parent ... is who I am". Without it I am nothing. Before the meme came along, people would say "Musician is what I am, or "A better person is what I am".

The meme is, in other words part of the contemporary identity culture. This meme asserts a person's right to the things that are most important to them: things neither the state not society have the right to take away. This may involve personal qualities, possession, avocations or group memberships, national, ethnic, religious, gender and so on). Such assertions are, to be sure, likely to be challenged by most people at certain points: people are unlikely to find many accepting an assertion that a safe cracker, multi billionaire, serial killer or prime minister of the UK is "who they are". It raises questions, much debated, about whether a people can claim any kind of right to anything they lay claim to as "who I am". But this is not a matter of English Language Usage.

Aside from that, the expression (meme) succeeds by being arresting. And part of that arresting quality has to do with a (let me call it) grammatically stretched use of the relative or interrogative who to cover "what kind of".

In the two examples of the question, "who" is a relative pronoun.   CaGEL's "interrogative" examples are wrong.   Those sentences are simple predicate nominative constructions, with "who" as the head of the noun phrase in the predicate.