If you remove the shelves a day and a night lamp should work fine. The heat lamp will heat the floor underneath it and the air, but your snake will be in his hide, not necessarily sitting on top of it where the lamp has heated. This is not an issue as long as your ambients are high enough, I have no experience using only ambient temps, I use belly heat so someone else will have to provide a bit more on that. Using lamps you will have to keep an eye on humidity, preferably between 60-80% but not so humid there is condensation.
Is your heat mat currently inside the enclosure? If so, I would unplug it as your reptile should never be in direct contact with the heat source.
You said it was a bearded dragon enclosure, I assume it would be sealed but you should make sure it can withstand high humidity without molding because beardies don't usually have nearly as humid of enclosures.