The corpora I checked indicate that both forms are used on both sides of the Atlantic. The BYU-BNC British National Corpus has 32 instances of chaperon and 32 of chaperone from the 1980s to 1993. The Corpus of Contemporary American English (COCA) has 277 instances of chaperone and 60 instances of chaperon from 1990 to 2015. (I excluded the spoken sections.) So there are clearly some differences across time and space, but chaperon is actually older or more British or both; it's definitely not a new American simplified spelling.
Surprisingly to me, the Oxford English dictionary treats "chaperone" as merely a common error for "chaperon", not even listing it alongside the other variant spellings such as chapperoon, shaparoon, shaparowne. It gets a special note:
[In the sense of a companion]... English writers often erroneously
spell it chaperone, apparently under the supposition that it requires
a feminine termination
If this is the actual origin of the spelling, I guess it would make it doubly erroneous now, since in modern usage the word is often gender-neutral.
But it's not clear that the e was intended to be a feminine termination. There are other words from French that don't refer to female people, but have definitely have gained an e, such as morale, locale, ladrone (the last one seems to be also taken from Spanish ladrón). An "extra" e also occurs in some words taken from other languages such as German (chorale, ketone). For most of these words, it seems to be connected to the stress or vowel quality of the last syllable. All the dictionaries I've found say that the primary stress in chaperon(e) falls on the first syllable, but some transcribe secondary stress on the third, and all seem to agree that it has an unreduced vowel (the Oxford English Dictionary gives the following three pronunciations: /ˈʃapərɒn/ /ˈʃapərɔːn/ /ˈʃapərəʊn/). This differs for example from apron /ˈeɪprən/ which had the same suffix in French but was borrowed earlier and now has a reduced schwa vowel (or a syllabic nasal) in English.
Another possible source of confusion is that the Italian augmentative suffix -one is spelled with an e. So English words taken from Italian often are spelled with a final e, which in many words is pronounced as a separate vowel (such as in minestrone) but in some words is silent, or optionally silent (trombone, provolone, calzone). (And in balcony, it was respelled to -ony).