Substitutes for beer... in beer cheese soup

Note: This is a suggested experiment, rather than a tried-and-tested solution

From the comments, yeast and malt extracts were suggested. I'd go further and say both, as neither will replicate beer flavour alone.

Both are rather variable. Some experimenting will be required.

In case you go too far with either of them, I suggest you have spare ingredients on hand to allow dilution (leftover soup should freeze well, to avoid waste)

I suspect that too much yeast extract will be worse than too much malt extract, so use caution on that. Yeast extract (referring to Marmite/Vegemite or own brand equivalents) is used in some vegetarian recipes as a source of umami; it's useful in savoury dishes but can get overpowering. It also has salt, so you might need to reduce the salt in your recipe.

I'd guess at a starting point of 2 tbsp of the liquid malt extract I've used and 1 tsp of yeast extract plus 0.5l (1 pint) of water for 1 pint of beer. That balance could be way off, but luckily both are highly processed and won't change much in cooking so you can add more to taste (as you can with salt).

A little acid wouldn't hurt (ideally lactic acid), maybe 1/2-1tsp of malt vinegar in that same substitute pint.

A way-out idea I'd be tempted to try (as I've got it on hand) is replacing the flour with dried sourdough starter, to bring in some yeast flavours and acid

Luckily that recipe doesn't call for a particularly bitter beer, as that's harder to get right. I wouldn't worry about bittering agents. You could though: hop extracts are rather strong (used at a few ml per gallon of beer, though they're available. If you wanted to try them you'd need to dilute the extract first then add, otherwise you might find one drop too little, two drops too much.