No warrant, no way.....
You sure they were legit LEO?
With the rash of impersonators lately I would be very very aware of your surroundings and whom is near. No warrant or balking at getting one would be a red flag to me.
You screwed the pooch allowing them in without a warrant so now you are in a defensive position.
You will probably have a hard time proving your rights were violated as you willingly allowed them in the house, the bedroom and to nose around in your safe.
Get ahold of Dudley and if you are a GOA, NRA or SAF member be contacting them. The more resources you have the better.
Document everything ASAP while it is still fresh and contact a lawyer in the AM.
I think I would also be inquiring with the Chief of Police how this behavior/inquiry is appropriate.