Learn Italian: Non che sia ingrato, ma sono davvero sovraccarico. - Not that I'm ungrateful, but I am truly overloaded.

This is the core of the rejected cause structure you are currently learning. We use non che followed by the subjunctive (sia) to dismiss a potential but untrue reason. Then, we use ma followed by the indicative (sono) to state the actual, factual reality.

Since the subjunctive sia is the same for 'I', 'you', 'he', and 'she', how do we know the subject is 'I'?

The second clause clarifies it by using sono, which strictly means 'I am'. Because the actual reason applies to the speaker, we naturally assume the dismissed reason does too. If you needed to clarify a different subject for the first part, you would add a pronoun: Non che lui sia ingrato (Not that he is ungrateful).

Can sovraccarico be used for mental stress, or just physical weight?

It works perfectly for both, much like 'overloaded' in English. You can use it for a physical object like a truck carrying too much cargo, or, as in this sentence, for a person who has way too much work or responsibility.

Could I use non perché instead of non che here?

Yes, but the grammar and tone shift slightly. Non perché is usually followed by the indicative (Non perché sono ingrato) and sounds a bit more direct. Using non che with the subjunctive is a softer, more elegant way to gently head off a misunderstanding.