Taking with some people on another forum they have given the following ideas to test it out.

The initial thinking is like you that there could be a problem with the rom daughter board.

However you can also rule out if there is an issue with the main Amiga A1000 motherboard. If you can find a space Amiga A500 or A1000. A500's can be picked up very cheaply these days for around £10/$10. You will also need some Amiga floppy disks.

I'm assuming the arcade machine's monitor is connected to the Amiga's RGB port? If so we can use this as it is to test. First disconnect the daughterboard. Now take the floppy drive and cable from the A500 and connect it up to the A1000 motherboard. And finally boot the A1000 motherboard. If it boots correctly you should be seeing the Kickstart rom boot screen asking for a disk to be inserted. To fully test you will need some Amiga floppy games. Once you have a couple boot these up and see what happens. If you still get graphics corruption then it is a problem with the motherboard and not the daughter board. If however everything is OK then it is the daughterboard.

How does the daughterboard connect to the Amiga motherboard? Parallel port? If so you can try an A500 motherboard instead with it, but you will need a gender changer to get it to plug in.

Hope that helps.

If you need any Amiga floppy disks for testing let me know and I can do you a couple of copies and post them to you.