You're going to have to free up some blocks on that PSOne card. As you have already discovered PS2 memory cards can only be used to save PS2 games and PSOne cards for PSOne games.
Fire up the PS2 with no game in it and go into the browser, then select the memory card. You'll see what games are saved on the card along with how many blocks each one is taking- if there is a particular game that isn't being played any more you can delete that to make room.
One other option would be to copy a save for a particular game onto the PS2 card- while it can't save directly you can move PSOne saves onto PS2 cards for temporary storage and just move them back onto the PSOne card when needed.