Warning: The following explanation is pure speculation: I am not a linguist, just a French with an amateur interest in linguistics.
In French, fête was written feste (or feſte) before the 18th century, and its /ɛ/ was¹ long to compensate for the lost s (see this wikipedia page). So Its pronunciation at time of its arrival in English was probably /fɛːt(ə)/.
Then, if one looks at Wikipdia's article on the Great Vowel Shift, one sees a transition in English from /ɛː/ (17th century) to /eː/ (18th century), then /eɪ/ (from 19th century). This would perfectly explain the modern (English) pronunciation of fête as /feɪt/.
By the way, as a native French, I'm really used to hear the native English speakers pronounce French /ɛ/ as /eɪ/, both in borrowed words and when they speak French. To me, it is part as the English accent as strongly as /ɹ/ for /ʁ/. So the short answer is maybe : because they say a French word with an English accent [insert Monty Python Holy Grail jokes here ;-)]
¹: Some French speakers still distinguish between /ɛː/ and /ɛ/, but it is no longer part of the standard French pronunciation.