So what do you expect the dowels to do in this instance? They have no compressive strength that will hold the box together like screws. They only have sheer strength the way you are installing them, and by removing 5/16 out of the MDF you are weakening it. As others have stated, the proper screws, installed properly, will be the best solution.
But it's your box, and it seems as though you have convinced yourself to use dowels, so I'm not really sure why you asked for advice?
There's no way I will use screws on the edge of a MDF board, I have made several tests, and found that it splits very easily, however I found that machine screws work better if you clamp the board while screwing the screw, but although it did not split it, I won't trust it won't eventually split it with a hard movement.
Take a look at this picture a took some minutes ago.
A - I screwed a #12 1.5" machine screw to a glued sandwich, it did not split although the screw went in so tight, not only because I pre drilled, but because it's very hard to split MDF on its face. To mention that I was near to the edge, but as I ran perpendicular to the fibers, nothing bad happened.
B - I attacked the edge with same screw used on A, I properly pre drilled and clamp the area with a clamp and two wood pieces, once I removed the clamp, it did not split, but just as I said before, I won't trust a screw inserted in that side.
C - Here I used a coarse wood screw, which even had a self drilling in the tip, anyway I pre drill with my drill before screwing. It split and ruined the mdf very easi and horrible, maybe because of that head type, that make even easier splitting.
D - Same test as C, just clampling the area with 2 more boards and C clamp, the damage was minor, but it split the edge.
You see guys? I think that you can use screws on mdf, but not in a edge. Dowel with get its compression once the glue expands and dries.