Is the signature of a letter a subject or an object?

The default pronoun to use in English is the objective case. See this EL&U.SE answer. For example, if you were to label a picture, you would label it "me at the beach in 2011" and not "I at the beach in 2011".

The signature is neither a subject nor an object, as it is not part of a sentence. Thus, the correct pronoun is "me".

I don't think signing a letter with a personal pronoun fits into the conventional format of a letter. As such, I don't think this question is really answerable. You can do whatever you want; you're already breaking the rules of letter-writing.

Some old-fashioned closings for letters made use of a copulative verb before the signature. E.g. see the following "Formal addresses and closings" from this web page:

To the Pope:
Formal Closing: On my knees before Your Holiness, protesting my filial dedication and imploring the favor of an apostolic blessing, I have the honor to be,
The humble and obedient Servant of Your Holiness,
(Signature)

To a person of high station or stature:
Formal Closing: I ask Your Excellency (Your Honor) to accept my profound respect. I remain
Your humble and obedient servant,
Or, I have the honor to remain,
Yours faithfully,
(Signature)

("Writing Letters – Small Manual of Civility", by Marian T. Horvat, Ph.D.)

In a sentence like this this, it seems that the supposedly "correct" pronoun would be reflexive, e.g. "I am myself" or "I remain myself".