If it breaks just remember it's only PS2

Now for a real answer swap magic seems like the best choice I did a wiki on it and 3.6 seems to be the best since 3.8 has some issue.

The con I see of this method is maybe some strain/ Wear and tear on the disk tray could be problem letter down the road and can be annoying if you play any Monster Rancher games you know what I mean.

There one up side to a mod chip It avoids all the cons of disk alternative and supplements it with it's own cons. Some chips are easier to install then other(not sure if there a solder-less one) All Voids warranty, I here the solder-less ones fall off over time and you can do irreversible harm. (and if suck a soldering even more so)

I say swap magic. Unless you don't case about the PS2 then go to town.

The easiest way to do it would be to get Swap Magic and install a flip top cover on your PS2. That's why I did and it was a very easy process. I never modded any system before but this was as easy as using a screw driver to unscrew a couple of screws and then screwing on a flip cover. Load the swap magic disk and then run the program, swapping it out with your game/burned copy of your game. It's as easy as that. I've had it for years and my slimline PS2 still works without hiccups.

I here the solder-less ones fall off over time and you can do irreversible harm.

ugh this happened to me on my first xbox. I was playing a game and it just up and came off. my xbox gave me some error and red ringed on me. even after opening it up and pulling the chip back in my xbox wouldn't boot. So I took it out and....er same problem.
I know its an xbox and not a ps2 but just be wary with those solderless pieces. like the rest say swap magics the way to go. I myself am ordering a modded ps2 right now, should be good.