Yeah, Engi is shortest. Engie sometimes also acceptable. Engineer when you want to sound intelligent. Engey is just UNNACCEPTABLE however, that's just ridiculous. That's like spelling Scout "Skawt" and Soldier "Soljr," you're just making stuff up!
You don't see a lot of English nouns ending in -i, so I spell it "engie." Ending it with -y makes it look like it's intended to be an adjective ("I'm feeling kind of engy today").
Consensus Via Google Hits
tf2 "engy" : 23000
tf2 "engie" : 22000
tf2 "engi" : 16000
tf2 "engey" : 200