It can probably run at 12v since I expect it has a switching regulator inside. But it may not be working as well as it should. You may be risking blowing up the computer's internal regs.
As noted, this is a crapload of heat for a linear reg. This is basically impractical to run this way. It would take a huge power transistor, heatsink, and makes a ton of heat. The alternative of a switcher is often a tricky beast but it's a more reasonable solution.
IMHO this whole project is not practical. Just get a new supply on eBay? It will be far cheaper, safer, and better packaged. There's just too much risk of blowing the laptop and laptops usually need easily portable supplies. If it's a new computer you certainly don't want to damage it, if it's ancient then eBay will have those supplies for next to nothing.