John,
Congrats on tackling the project of building your own router plane.
The only router plane that I have built is the one that I posted so I do not have any direct experience with the problem you are having. I have looked on line at many of the router plane designs, and have looked at the “eyebolt” type of design – which I think is what you are building.
Here are my thoughts on what could be some causes for what you are experiencing:

There are only two reasons that I can think of for why the cutter is not staying set:
1. There is not enough clamping force supplied by the eyebolt
2. There is enough clamping force supplied by the eyebolt but when the plane is being used, the materials that are holding the cutter in place are Yielding (permanently deforming due to the stresses applied). Once any yielding occurs, and the yielding can be really small, the clamping force is drastically reduced so the cutter will move.

My gut feeling it to start looking at the surfaces that contact the veritas cutter (the long post) to hold it in place. Either the eyebolt or the notch that you cut into the plane. Look for signs of yielding due to use. Since the cutter is presented at a downward angle to the wood being planed, and since the eyebolt is above the cutter… the bottom of the post is being pushed into the very bottom surface of the notch that you cut.
Im not sure if I am explaining this very well, and I can send a diagram of what I think is happening if you need me to.
To put it another way, the post is trying to rotate around the eyebolt and pushes hard against the very bottom of the notch you cut. That will be a high stress area in use.

Now if that notch is not uniform and smooth, and fitted well to the Veritas post the large forces exerted on the small surfaces of the beech that are contacting the Veritas post will yield. Large force over a small area equates to high stresses which most likely exceed the strengths of the beech.

So,
All that being said,
Here is what I would do:

If you find signs of yielding as I think you will (it will look like bruising on the beech)
cut the notch for the Veritas post deeper. Go to the Home Depot and get some 1/8 or 1/16 inch aluminum angle that is the size of your choosing (1/4 inch legs would work). Use JB Quick to bond the angle into the notch you cut. When you glue it in place place the Veritas cutter in it and tighten it with the eyebolt so it is seated exactly as it will be in use. The aluminum will provide a consistent surface for the Veritas post to clamp against and will most likely take care of the yielding you are experiencing.

I hope my thoughts help you. Im in no way an expert on any of this.
I wish you the best of luck with your project and would love to know what the final solution is to the problem you are experiencing.

Rog

What blade did you use? I’m thinking I like the design of your plane and might try to build one with a 1/2″ spear point. Would you advise against that one? Not sure exactly what I’ll be doing with it but mostly smoothing things like the bench leg recesses and other odd jobs. I don’t know that I’ll do much housing rebates for shelves and such so the 1/2″ width shouldn’t be a problem. Your thoughts?