Apparently, you killed the diode on the input (within the chip, so it makes a short-circuit between Vcc and Gnd, that's why it heats now)
Technically, you can replace the SAM3X, but not sure it will be valuable in terms of economic balance. A new SAM3X costs already 11 euros from Mouser, without the packaging costs. And you will need to buy some extra stuff for the soldering (expecting you have a thin soldering iron)
You can replace a LQFP package without an hot air iron, but you need a very thin tip on your soldering iron. Personnally, I use 0.8mm Antex tip for that. The second thing that you will need is unsoldering braid and soldering flux (if possible, the "no clean" type). You will also need a small cutting plier (with the smallest possible extremity, to work near the component body). This will be probably the most expensive tool you need to buy
The flux is sold in various formats, the most useful is the pen model. DO NOT TRY TO WORK WITHOUT FLUX! With such compact packages, you have no chance to avoid solder bridges between pins, and that's where the flux comes into the game.
First you will need to cut all the pins around the SAM3X, in order to remove the body. For that, use the micro cutting plier to cut the pin at body level. Take care not to bend the pins too much, otherwise they will pull on the copper track and there is a risk to damage it.
Once the body is removed, use the iron to remove all the pins, one by one. This is rather quick to do, just use the soldering iron for that.
Next step is absolutely mandatory : you must clean the pads where the chip was soldered. To do that, use the flux, spread it over the 144 pads. The flux will help the solder to melt and clean the pads. Use the soldering iron to melt the solder on the PCB and use the unsoldering braid to remove the old solder. Take care not to overheat the PCB. The best thing is to clean 5 or 6 pads, then let the PCB cool down for a few seconds before you do the next ones
Once this is done correctly. you shall have a perfectly flat PCB where the SAM3X was soldered. If not, then you will have the biggest problem to solder the new chip, because all the 144 pins must be aligned on the same plane.
Clean the PCB to remove the flux with solvent (the flux turns brown and becomes sticky once the soldering iron has used it)
Put some clean flux on all the pads to keep them clean for the next step.
Put the new SAM3X on the PCB, and make a tiny solder point on two opposite pins. This will hold the chip while you solder all the other pins.
Apply flux on the pins of the SAM3X. Do not hesitate to "moisten" the PCB. It's better to have more than not enough.
The next step is also critical and shall be done as fast as possible. You will need to solder all pins by moving the iron. It shall be slow enough to let the solder go on the pin but fast enough not to heat the chip too much. A good method is to make one side, let chip cool down, second side, cool down, etc.
During this phase, make sure that you put enough tin on the pins. VERY IMPORTANT : it does not matter if you make bridges between pins. It's better than having not enough tin and make a bad soldering joint.
And last step : using the unsoldering braid, remove the solder in excess. With the flux, the solder will go between the pin and the PCB pad, and that's where it is important. The braid will remove what is in excess, not what is under the chip.
And you are done....
As you see, it's possible to replace the chip, but you will need tooling that is not so common with amateurs and rather expensive. And you need to get the "touch" to do that. It's not hard (I am soldering 0.3mm pitch chips on prototypes following this method) but there is a serious risk to fail the first time.
So.... good luck