You don't need to cut the aquarium to make an overflow!!!!!!
some pvc pipe and elbows are about all you need. I like 20-25mm thin wall pvc for the purpose since the smaller stuff couldn't keep up very well and made me worry about an accident involving wet floor.
I'm not finding a good picture of a complete pvc type no holes overflow but here are a couple pictures from my thread.


It's basically 5 bits of pvc along with 6 elbows.
one bit of pvc is long enough to go across the top edge of the aquarium over to the grow bed. the next two bits of pvc are the same length and they dictate the high water level in the fish tank. The last two bits of pvc are only long enough to attach two elbows together. I've never glued my no holes overflows. They work on the same principal as a water bridge really. You put it together in roughly the shape you want and then submerge it and flip it all about to get all air bubbles out of it (good idea to do this every so often as air bubbles can build up in it and stop it working right.) Then carefully keeping it level, lift out of water and set in place. Then test it out.
Do be ware that small fish can swim though a no holes overflow so you want a screen or net protecting it so fish can't enter. I did have a 2" or more fish swim through a no holes overflow made of 3/4" pvc and wind up in the grow bed bucket.
As for what type of grow bed is best. there are many opinions about that. Generally 30 cm or 12 inches is a standard good depth though people have made shallower and deeper grow beds. Something like a feed trough might work well. My small indoor system uses 5, 5 gallon food grade buckets as well as a few smaller ones. I've made plenty of wood grow beds and lined them with pond liner. They seem to be working very well in the outdoor system. There are many ways to do it.