The Consul is a classic example of an anti-hero, according to the traditional definition. He lacks heroic qualities such as a moral code, altruist tendencies, and courage; despite this, he is the central character of the novel around which all events and characters revolve. According to Gale Cengage, anti-heroes:
...typically distrust conventional values and are unable to commit themselves to any ideals. They generally feel helpless in a world over which they have no control.
This perfectly describes the Consul, who feels fated to his failures due to a deterministic universe. In his eyes, he cannot take any individual steps to change his fate, and so he embraces it through his alcoholism and depression. His refusal to take action shows his failure as an individual; he has convinced himself that he is destined for Hell, so he sees no purpose in attempting to change himself or his situation.
Even though here was God's moment, the chance to agree, to produce the card, to change everything, or there was but a moment left... Too late. The Consul had controlled his tongue.
(Lowry, Under the Volcano, Google Books)
The Consul here explicitly refuses to "change everything" by holding back important information. He does this because he inherently believes himself to be unworthy of good things; his life is worthless and so he has no right to try and better himself or his situations. Without that ambition to grow and change, the Consul cannot be seen as anything but an anti-hero, and in fact is almost an antagonist.
Note: the modern use of anti-hero increasingly refers to a protagonist who is unlikeable but still does the right thing in the end; many modern protagonists are intentionally unsympathetic so as to provide a character arc. The Consul would not be seen as an anti-hero under this cultural definition.