A few problems with this situation.
First off…. The Police will ‘not’ attempt to ‘find the owner’. They have no inclination to do that. And that is not one of their functions. They would have no better chance of locating the owner than you.
Many jurisdictions can have different policies, but often the money can be claimed by the finder if the ‘loser’ does not claim the money after a certain passing of time.
The Police saying if the owner is not found, the money will be filed as unclaimed property is Pure Horse Puckey.
What logical or ethical reason would justify that unclaimed money would have a better destination in unclaimed Property Locker than in your possession?
You should feel no guilt for finding money.
You could have sat on it and kept alert for anybody asking around or posting in a variety of places about losing money. The person asking would be the only person knowing the exact amount besides you.
If a month goes by without any leads, either keep the money or if you feel ‘funny’ about it, donate it to a local Children’s Hospital or Soup Kitchen serving homeless people.
You should go down to the Police Station that sent the Officer to your house. Hopefully, you got the Officers’ name/badge number? Tell them you felt you did the right thing by turning the money in, but that if the money isn’t claimed, you feel you should be entitled(as the finder) to keep the money.
Ask them what is the Law in their Jurisdiction?
They should ‘know the Protocol’ concerning lost/found money. And…. Because you were such an ‘Honest citizen’ for turning it in, they should have no problem telling you the Law.
….If they tell you the Protocol without hesitation, you will then know…. The Law.
If they decline to answer your questions…. If they say they are too busy…… >>>
Then, at the very least, ask them to invite you to their next Department Barbecue.
Because, you paid $1300 to Fund their party.