Whether or not the heat pad will be sufficient to warm the tank will depend on how warm or cold the room is that the tank sits in. It will certainly warm up the floor of the tank to where you need, it, but it may or may not keep the "cool" side warm enough on its own. If not, you'll want to continue with the supplemental lamp.
The heating pad you would buy in a petstore does NOT come with any sort of control, and it MUST have something, otherwise it can/will get too hot and risk burning your snake. You should get a good thermostat for it, or at the very least, go to the hardware store and get a rheostat (lamp-dimmer switch). They come already wired and ready to just plug in...very simple.
To use a CHE, you don't need that whole wire-mesh contraption. You can put it in a lamp dome....however...it must be a ceramic dome rated for high wattage....which is also what you should be using for higher-wattage bulbs. A standard house lamp, or cheap dome will not work (as you've already found out.) You should be able to find a ceramic dome (meaning the light fixture inside is ceramic...the outside still looks like cheap aluminum or black metal) at your pet store. That can sit on top of the screen...just make sure its big enough to hold the heat source up away from the screen so its not touching.