I don't think it's an exception so much as a way of pronouncing as /eXe/ the end of certain words that can also be pronounced as /ɛXe/, where X stands for some consonant sound.
The same pronunciation is observed, for example, for all the verbs in -êter that I've found in wiktionary, such as
Prêter / Apprêter
In all of these cases, the pronunciation of the end of the word can be /ete/ or /ɛte/; both will be understood just as well. Frequencies of each option may vary between words and regions, but I don't have any data about that. The same flexibility in pronunciation options exists for other conjugations that end in /te/, such as the second person plural indicative present (vous prêtez) or the participe passé (j'ai enquêté)
In all of these cases, /ɛ/ must be used in conjugations that end with /tə/, such as the singular present indicative. Using /e/ would be characterized as a mispronunciation, e.g. to pronounce the ê in il m'embête or j'arrête, one must use /ɛ/, not /e/.
I'm not sure about cases like "nous arrêtons, vous enquêtiez". My sense is that /e/ is OK here too.
Finally, it's not specific to verbs ending in -êter or -êcher, the same possibility of sliding from /ɛXe/ to /eXe/ happens also at the end of words like aimer, baisser, mêler, laisser, errer, etc.
(Thanks Gilles ♦ for helping me see that it is a more general phenomenon than my first response.)