Poor man’s router with wedge fail

I tried making a Paul Seller’s style Poor Man’s Router plane. It worked quite well but I couldn’t get the chisel’s blade edge to be really perpendicular to the surface of the wood. As an alternative I thought I’d try cutting a slot into the piece of timber and making a wedge to fit it to hold the chisel in place.
 

As much as I tried I couldn’t get the wedge to hold the chisel firmly enough that it wouldn’t slide up once I started cutting into wood with it. I think this was partly to do with it being a bevel edge chisel. This meant there was less surface area in contact with the wood, so less friction and easier to slip.
I might try making a hardwood wedge for it. I also need to try a firmer chisel instead of a bevel edge one. The extra surface area might be enough to hold it.