I already have a modded PS2, but the PS1 is what I need.
What type of PS2 mod do you have?
Quite a lot of PS2 mods work perfectly fine for playback of PS1 games, it's just that often some of the people providing these mods aren't very good at informing the buyers about it.
If it's a swap trick mod of some sort (using things such as Swap Magic) you can actually just get a PlayStation gamebooster/Action Replay/GameShark and that disc will work perfectly fine as a swap disc for PS1 games. Just boot the disc that will act as a PS1 swap disc, then wait 'til you hit the menu and then perform the swap and select Start Game.
If it's a mod chip, then you could try out some of these methods:
1: Some modern mod-chips have proper auto detection for inserted media, so just try to boot the game as any other game.
or
2: If your PS2 auto boots PS2 imports, but evt. requires some button to be pressed to play DVD movies. Then try to set the PS2 in standby mode after inserting the PS1 game, then hold reset pressed 'til the blue light turns on (roughly 2 seconds) and see if the game boots. (Or evt. try to use the same method you would use to start DVD movies on it)
or
3: If your PS2 requires a button to be held on boot-up for imports to start (Like holding O pressed or something), just try to hold some of the other buttons pressed. Chances are that one of them will indeed boot your PS1 game.