Assuming you are short straddles/long Flys,you should ask yourself why you are delta hedging.
If you run a short vol/gamma book one is obviously delta hedging so the 2 standard + move doesnt send you the way of the dinosaur.
The other reason is to hopefully extract the differential between implied vol and "realised" vol.
Obviously,if you sell a straddle at 24 Vol,you feel the realised vol should be much lower.Its a pure bet,and if you are correct,you will be rewarded.With a butterfly,your greeks are tame relative to the straddle,hence its not as pure a vol play as the short straddle.
You can certainly delta hedge a fly,but in order to bank the same coin,you will have to trade much larger.It really comes down to your comfort levels.Would you rather be short 1 straddle or long 5 flys(approximation)..
I personally dont sell straadles in this VIX enviorment,but I do trade Flys
I have seen an article where backtesting shows that delta hedged short straddles are profitable in the long run. I wondered if anyone has experience trading butterflies and delta hedging at the end of each day, and if that is profitable?