Earlier quoted context omitted.
Language, and especially its mechanics like grammar, are entirely a distraction w.r.t. "cogito ergo sum". The underlying argument it points to is language-independent.
Correct. Here is the stub of a reply I can't be assed to finish right now: Words and language refer to sensations (P.I. §244: "How do words refer to sensations?"). Sensations can exist independently of language to refer to them (P.I. §256: "—But suppose I didn’t have any natural expression for the sensation, but only had the sensation?"). Thus it can be possible for one to experience the cogito, the mere act of aware…
"I think therefore I am" is invalid, and what's wrong with it is the "therefore," the idea that you knew one thing, and you drew a "logical" conclusion from it, in a "prior to language" environment where words have no meaning, where "true" and "false" are indistinguishable, and logic is impossible.
Logic requires words. "Logical" means "verbal," from the Greek logos (λόγος). You can't have a logical argument (you can't draw a conclusion) from the instantaneous standpoint of someone "experiencing" cogito, where words mean whatever you want, or nothing at all.
The experience you're having is not a logical argument. As a sentence, "cogito ergo sum" is invalidated as soon as you write it down in a shared language.
I'm sure it feels right to you! But you can't actually say anything true about it in English, or Latin, or any other shared language.
Thereof, you simply must remain silent.