Doubt about the subject in this phrase: I, me, or myself?

It is as straightforward as this: in English, prepositions never take nominative case (which is what "I" is). Since the pronoun in question is modified by "with" (a preposition), the correct case is definitely accusative "me".

Furthermore, if two or more phrases are connected with a conjunction (like "me and a pleasant lady"), and all of those are treated as one unit that the preposition modifies (as is the case in your example), then every conjoined element in that unit gets the same case distributed throughout. So, with or without the "pleasant lady", the pronoun should be "me".

The use of "myself" is a somewhat recent addition to English. It is likely that this sprang out of people's generally uneasiness in using "me" in any conjoined phrase, because of learning enough about pronouns to know that "me and John are going to the park" is informal and poor English, but not enough to be sure when "me" is okay to use. So "myself" is a way to avoid using "me". This avoidance has become common enough that "myself" is on the verge of being a standard alternate way to express this. (But from a linguistic standpoint, it is quite odd!)

A plausible alternative could be rendered "...and myself." I, personally, would render it as:

I was left alone in the pub at the end of the night with a cheerful and pleasant lady whom I had met during the last minutes of the meeting.

Part of the problem is that the clause after "with" is not complete. A shorter sentence makes it more obvious: "The house was empty, with only a dog." With only a dog what? I'd re-write is as:

At the end of the evening, the bar was almost empty, with only me and a very cheerful and pleasant lady I met in the last minutes of the evening remaining.