Can you introduce a direct quotation with "that" and not accommodate conjugation and pronouns?

I've heard this many times recently in public, copy-edited discourse, where I don't expect to find grammatical errors, but it doesn't sound right to me. Let's say my mom says to me, "I want you to clean your room." I can report what she said with a direct quotation:

My mom said, "I want you to clean your room."

Or I can report it indirectly:

My mom said that she wanted me to clean my room.

My question is, can you introduce a direct quote with "that" without accommodating it? e.g.,

My mom said that "I want you to clean your room."

Is this acceptable? This sounds wrong to me since "that" makes it sound indirect, and that I, the speaker (not my mom), want my listener (not me) to clean their room. I thought it would need to be like this to be correct:

My mom said that "[she] want[ed] me to clean [my] room."

(of course, the brevity of this example quote hardly justifies this, but ignore that)

Solution 1:

'That' introduces a subordinate clause, and more specifically in instances such as 'she knew that' or 'I believed that' or 'he thought that' introduces indirect discourse.

Indirect discourse has its rules such as the shifting of tense. For example direct: She said, "I am ready."
indirect: She said that she was ready.
present tense --> past tense
direct: John said, "I will go."
indirect: John said (that) he would go.
future tense --> conditional tense

So, no. In the last example, it is impossible to say: John said that "I will go."

Some links for indirect discourse:
https://www.thoughtco.com/what-is-indirect-speech-1691058
https://www.frontiersin.org/articles/10.3389/fcomm.2020.606616/full