There shouldn't be any problem. Corruption is a common problem though, as in all latin american countries (I lived in Punta Cana for 8 years). Don't let anybody at the airport tell you you have to pay extra for anything.
I would recommend disassembling the detector and pack it in the hand luggage.
There are some videos on youtube about metal detecting at Punta Cana.
Every resort has security guards on the beach. Don't be affraid, they are there to protect the tourists from too many vendors, prostitutes and criminals. You're perfectly safe there, don't worry.
There are lots of watersports activities and boats can travel only in a relatively narrow channel between beach and reef. So there is a lot of boat traffic close to the beach during the day. Stay within the marked swimming areas. In the evenings around 5 pm most boats will be taken out of the water. Then it could be worth a try detecting the part of the beach where people get on and off the boats during the day (usually that's right in front of the watersports center). They lose everything you could imagine. The dominican beach boys and hotel staff are curious but friendly, you might become friends with some of them. Especially the guys working in the watersports centers usually know good spots where to find lost things in the water or where a tourist had lost jewelry.
Most of them know a canadian detectorist who's been living there for years.
Be careful with the stingrays. They are everywhere, even in the shallow water and swim zones. Don't step on them.
Some people might tell you they saw a sea snake. There are no sea snakes but snake eels. Completely harmless.
Have a good trip and enjoy your vacation! Looking forward to see the posts with your finds. Good luck and happy hunting
Chris