A towel won't help much at all, just like blankets won't help raise their body temp. Cold-blooded reptiles don't work that way unfortunately. And you cannot use any type of "night bulb" that emits any color of light, as it will totally disrupt his sleep. His tank must remain pitch-black at nighttime.
As already mentioned, if his tank temperature drops below 65 degrees at night, as you said it does drop to 60 degrees (not too bad at all actually, 60 degrees is very close to an acceptable nighttime temperature for a Bearded Dragon, as you have to remember that they naturally come from the Australian Desert, which is extremely cold at night, which is the way they like it), you need to buy a Ceramic Heat Emitter (CHE), which is a "bulb" of sorts that emits no light at all but only heat.
However, YOU CANNOT USE A 100 WATT CHE!!!! As Venus recommended a CHE she was correct, but remember, you are only trying to bump his nighttime temperature up from 60 degrees to between 65-70 degrees max, you do not want his tank any warmer than that at nighttime, as they only sleep comfortably and soundly at night if their tanks are pitch-black and considerably cooler than their daytime temperatures, and 65 degrees is an acceptable, comfortable temperature for a Bearded Dragon's tank at nighttime. If you were to use a 100 watt CHE or any wattage even close to that, his nighttime temperature would skyrocket to being close to his daytime basking spot surface temps!!! This would not be good, and he would be miserable at night and would not sleep well at all, which will result in an array of health issues.
my lizards all enjoy a room temperature never lower than 22 degC , and their warmed hides , see above post, my beardies have the option to use these , set at 36 degC .
As you're only trying to bump his nighttime temperature up between 5-10 degrees, you only need to buy or order online a CHE that is at most 10-20 watts (easier to find different wattages of CHE's online, as well as much, much cheaper, Amazon.com and any type of online lightbulb stores will be cheapest). Please do not put a CHE that is any higher of a wattage than 20 watts, as even that may be too hot. All CHE's fit into regular clamp lamps and dome lamps, and I suggest putting the lamp with the CHE directly over the place in his tank that he usually sleeps. If you're putting a CHE of 10 watts directly over him at night, and the Ambient (air) temperature of his tank is already 60 degrees, which is almost a perfect nighttime temperature for a bearded dragon and he almost doesn't need any nighttime heat source at all, then he'll be just fine. As I said, even if you buy a 20 watt CHE and put it directly over where he sleeps at night, or at least over the side of the tank that he typically sleeps at night, and it bumps his tank temperature from 60 degrees to 75 degrees, that's fine. But you do not want his nighttime temperature to go any higher than 70-75 degrees, as if it gets up to 80 degrees then you're making his nighttime temperature the same as the daytime temperature of the Cool Side of his tank, and this won't be comfortable for him at all. So please make 20 watts the max.
I would just Google search "20 watt CHE bulb" and find the cheapest place online to order it from, Amazon.com is probably cheapest. You'll probably save at least 50% ordering it online than if you were to buy one in a store, specifically a pet store. There's no difference between a CHE from a pet shop and one from a hardware store like Lowes, Home Depot, Tractor Supply, Amazon.com, etc. And Amazon.com will have a lot more wattage options as well. I just wanted to make sure that you didn't buy a 100 watt CHE for him, as that will literally cook him at night...