How can I minimize "stringing" of cheddar cheese

Sodium citrate - https://www.amazingfoodmadeeasy.com/info/modernist-ingredients/more/sodium-citrate
You can find the ratio you think is best on that site. I would make a sauce from it. Super thick then chill and cut it. I think it would be best to use some tomato centers, puréed, with a touch of water for your dissolving liquid.

Use more mature cheese.

Cheddar runs the gamut from very young, mild, melts like mozzarella in strings, right up to so mature it has salt crystals in it & is crunchy, which barely melts at all.

Most supermarkets [at least in the UK] carry mild, medium, mature, extra mature & vintage. Specialist cheddars then run even further, to 'crunchy', or 'crackly'.

Try at least one if not two categories older. Vintage is probably too mature for a chilli & would be a tad grainy if you try to melt it.

In the UK - I don't know about anywhere else - there is a rough guide to cheese maturity marked on the pack. This is a simple & quite broad numbering system, from 1 being the mildest to 5 being the strongest. It is not necessarily directly related to the age of the cheese, but there is a reasonable correlation between the two.

From comments:
In the US aged cheddar is called 'sharp', the more aged it gets the more superlatives they add to the name, like super sharp, super extra sharp, etc

and… as we're getting some rather preposterous claims in comments about cheese ages, let me point out that cheese is not wine or whisky. You don't generally mature it for years unless you want something very specialist [& not coincidentally very expensive.]

UK Cheddar ages, approximately.
Mild is typically about 3 months of age, medium matured is 5 to 6 months old, mature is around 9 months, Extra Mature around 15 months and Vintage is usually 18 months or more.

Anything older than that is getting into specialist territory.