Yes, you can. You lose the statue quest after leaving Cookie's House, and the trigger for DoubleE to stop is holding cookie's note, which Cookie only drops before you finish quest 11.
You need to:
* Talk to DoubleE, get quest 11
* Talk to Cookie, let him drop the note but leave it
* Pay DoubleE, fix the statue
* Talk to DoubleE at night
* Leave Silk Church and pick up Cookie's Note
* Go to Cookie's house
This is definitely unintended (Tinta even says, "I doubt we can make friends with both"). You even get 2 screwdrivers this way.
Yipee I'm five months late!!