Another possible way to go is a large 12V sealed lead acid (SLA) battery and a trickle charger. This method also has it's drawbacks. And NEVER use a regular car battery indoors! They release gasses that can be explosive!

If you don't yet have a car amp I'm not sure I would persue this project either. One of the reasons car stereo amps are cheaper is they don't use a big 120V mains transformer/power supply. It's just not electrically efficient nor cost effective to convert AC to 12VDC to supply an amp that converts the power again. And no, the amp can not be easily modified to accept the 120VAC directly. You see, inside the amp the 12V power coming in is fed to a converter circuit that pumps-up the voltage to something like +/-25V or more depending on the power rating on the amp. If you are really determined to do this the best way would be to build a bipolar DC power supply so you could bypass the DC-DC converter and feed the amplifier circuits directly. This would give you a circuit architecture most like a typical home amplifier. But this would be a pretty advanced DIY project and if you aren't up to the task yourself getting someone else to build it for you shoots the budget again.

Using a bench power supply like I linked for you will work and is probably the cheapest (but not really cheap) way to go. Car amps draw tremendous amounts of current and you have to use a very large unit so you don't strangle the amplifier.

So like I said, unless you already have something you are trying to utilize I wouldn't go out and buy anything to go in this direction. You would be better off looking for a used amp designed for home use.