This should be staight forward as long as you have some good G clamps and can hold all the parts very securely.
1. First anneal (heat to cherry red and cool or quench) your rod to make it workable when cold. You are basically following the noble art of riveting.
2. Allow no less than one and a half (1.5) times the thickness of the bronze plate for peening (riveting) over. More if you want to have a raised head. Countersink the holes before peening if you want the peened heads smoother or slightly flush, you do need some raised head for strength.
3. Securely clamp the assembly together including the rod with a series of washers or a spacer to hold the rod the required extent while you peen the first side. You may need an assistant to hold a "dolly" against the rod on the opposite side while you do the riveting on your side.
4. Use a large ball pane (Peen) engineers hammer say 2lb and by using the ball gradually peen around the circumference. You will get the idea. Try a dummy run with some scrap first. Happy riveting, I have to go sailing......
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