An excellent question, although you may not find any answer completely satisfactory. This is purely my informed opinion as a native British English speaker, who also has a knowledge of French.
You already know it is a loanword from French, and the accent appears in the French word. So what you seem to be asking is why English speakers would continue to use the accent?
It is true that we don't have accents in written English. However, fiancé is not the only word you may see written with an accent. Café, for example, is written in English as often with an accent as it is without.
Accents are present in other languages as a pronunciation guide. I would argue that their use in English is not so much a guide, because a majority of English speakers worldwide likely do not know how to interpret French accents - rather, I would suggest it acts as a kind of pronunciation indicator, ie that it is a loanword and has a French pronunciation.
"Fiancé" is by no means an uncommon word in English, but it is not as widely used as "café" which I would argue most English speakers know how to pronounce - nobody is likely to say it to rhyme with safe!. "Fiancé" is used far less in comparison. Seeing it for the first time and without any reference, a native English speaker may pronounce it similarly to the word "finance"! But seeing the accent is a clear indicator that the word is of French origin and guides pronunciation - especially if you are familiar with cafés!
I would also suggest that accents are retained for styling purposes. It may be a generalisation, but the British tend to regard the French as having style, and certain things such as coffee and "coffee culture" as synonymous with France. Just as the flag of the USA is often used in advertising of typically North American things like burgers, jeans, cola etc, I would say that the word "cafe" styled as café on signage, menus etc in English may be done to suggest a level of "class". It may be a similar case with the words "fiancé" and "fiancée" - they tend to be used only formally in English, and as such the users may want to retain the gravitas.
As an interesting aside, the French do not insist on accents on some signage where block capitals are used, which suggests that fluent speakers do not always need a guide for words they can recognise at a glance. If we can recognise "cafe" without the accent and you are fine with "fiance" then perhaps accents in any language are, to a degree, archaic?