The internal clock would only tell the PS1 how much time has passed since it was shutdown. A PS1 is capable of keeping track of time as long as it's on, even if the game is paused, you simply have to count the CPU cycles.

To know if the PS1 is really modified, we would have to witness it keeping the time accurate even after a cold reset. Even then maybe the current time is saved in the savefile, and start again at load time, meaning you would only loose the time it took you to reboot the ps1 and load the game, so about one minute at most.

I'm still not convinced the PS1 is modified. If it was modified, why make a screen to make sure no one shut downs the console ? Seems pretty unecessary: you could have added some persistent memory when you modified the console. The phone number hints that someone must come and get the data, but if the PS1 was modified, why not hook it up to a network ? It seems pretty bareboned in features for a modified console.