Parsing a passage in Byron's Don Juan: what is the subject of "offer" here?

The whole block of poetry quoted in the original question is divided into four logically distinct sections by the three semicolons that Byron inserts in the lines. Thus we have a first section consisting of

"Man's love is of man's life a thing apart,
'T is woman's whole existence;

a second section consisting of

man may range
The court, camp, church, the vessel, and the mart;

a third section consisting of

Sword, gown, gain, glory, offer in exchange
Pride, fame, ambition, to fill up his heart,
And few there are whom these cannot estrange;

and a fourth section consisting of

Men have all these resources, we but one,
To love again, and be again undone.

The first section consists of a comparison of the status of love in man's life (where it is "a thing apart" from the rest of his life) and in woman's life (where it is everything). The second section notes that man may direct his attention (presumably after an episode of heartbreak) toward government, military affairs, the church, maritime pursuits, or the business world. The third section further discusses man's options by discussing what motivation impulse each career option may serve, and then observes that most men's attention to one or another of of these objectives estranges them (in a beneficial way) from other objects of attention (such as the women they love or had loved in vain). The fourth section returns to the comparison between men's many options for devotion and women's only option, and concludes that the lack of alternative opportunities and objectives causes broken-hearted women to seek meaning and direction in new love, which will again disappoint them.

As the comments by Jim, StoneyB, and Nigel J indicate, the sense of the third section is that the four options listed first ("sword [military prowess], gown [scholarship or religious study], gain [financial success], glory [ill defined here but presumably including renown for excellence in such rock-star fields as politics and poetry]" can feed man's desire for "pride, fame, ambition." Byron uses commas in three different ways in this three-line section of the quoted excerpt: to indicate parallel entries in a series (the commas in "Sword, gown, gain, glory" and "Pride, fame, ambition"; to signal a break between independent clauses (the comma after "heart"); and to mark the end of a series of parallel entries (the commas after "glory" and "ambition").

As StoneyB remarks in his comment above, this last use of commas is not standard according to current punctuation conventions—and it may not have been in Byron's time either, although he was writing near the end of an era of profligate comma use. In any event, if I wanted to express the three lines in question in accordance with standard modern use of commas and conjunctions (specifically the word and), I would write them something like this:

Sword, gown, gain, [and] glory offer in exchange
Pride, fame, [and] ambition, to fill up his heart,
And few there are whom these cannot estrange;

The reason that I didn't remove the comma after "ambition" in my revised version of the lines is that it performs another function besides indicating the end of the "Pride, fame, ambition" series—namely, separating "ambition" from "to fill up up his heart," to avoid the misreading that Byron intends "ambition to fill up his heart" is a unified phrase referring to a desire to fill up one's heart. As you can probably infer from this presentation, "Sword, gown, gain, [and] glory" is the subject of the first independent clause, and "offer" is the verb.

The twin omissions of "and" obscure Byron's intentions a bit, but StoneyB is surely correct that the poet axed them for metrical reasons and not because they interfered with his intended meaning.

The single most essential step in making sense of the passage, however, is to understand the force of the semicolons in establishing the four sections outlined above.

The subject of offer is

Sword, gown, gain, glory,

A list like that does not always have to be after the verb.

I'm curious why you think it might be man, which I assume you realise is in a different clause...