Can bacon or beans suggest some sort of alcohol?

The collocation cakes and ale goes back to Shakespeare's Twelfth Night, when Sir Toby Belch (drunk, as always) rounds on the sanctimonious courtier Malvolio with "Dost thou think, because thou art virtuous, there shall be no more cakes and ale?".

The Englishman first appeared in The Flying Inn (1914), some months after Nuts and Wine a theatrical revue with lyrics by C. H. Bovill and P. G. Wodehouse was staged in London. Walter Dendy Sadler's painting Over the Nuts and Wine (1889) was extensively reprinted as an icon of relaxed camaraderie among English gentlemen.

I'm not aware of any special connection between Englishmen and bacon and beans, but if you compare that chart with pork and beans it's obvious Brits prefer bacon where Americans go for pork. I suspect the English connotations for bacon and beans were more akin to Jeffrey Archer's Krug (champagne) and shepherd's pie (simple yet convivial fare), where to the average American [salt-]pork and beans would be dreary trail/frontier rations.

The first stanza isn't "about" food and beer, nor is the third stanza "about" food and wine. It's just that every stanza ends with a whimsical couplet involving two things that Chesterton's archetypal Englishman would see as a natural pairing.

EDIT: At the risk of steering into litcrit / social history territory, thanks to OP himself for ferreting out the crucial context. The Flying Inn (a novel interlaced with poems) is set in a future England where the Temperance movement has allowed a bizarre form of "Progressive" Islam to dominate. Chesterton is satirising the fact that from an Englishman's perspective, the natural pairings for cakes, beans, and nuts (i.e. - ale, bacon, and wine) would be prohibited in a Islamic state.

Neither is suggestive of beer. Instead, along with the other contrasts, this one is between meat and non-meat, saying he requires both.

It also gives a suitable rhyme, of course.

Sounds to me like the suggestion is that an english breakfast should always have bacon, and if you're going to provide beans then there'd better be bacon with them. This might sound bizarre but if I was served an English breakfast that didn't contain both, then as an Englishman (or partly at least) I wouldn't call it a full English breakfast.

In this case at least, I would not expect that either the bacon or beans reference alcohol, but are there purely to convey what constitutes a proper english breakfast.

This is really just a way of saying St. George is a proper Englishman. If it was about a German, it could be, don't give him any schnitzel, unless you give him `kraut.