I had the same issue happen with the traveler dead eye on my boat - after years of use, corrosion caused the holes in the crossbar to wallow out and the aft rivets became ineffective. I would offer an alternate solution to relocating the bracket. The other option is to drill through to the top surface of the crossbar and then thru-bolt the bracket. The advantage of repairing this way is that, first off, the bracket remains in the same location. More importantly, you don't have a cluster of eight holes in the bottom of the crossbar that are close together. An excessive amount of holes could potentially lead to damage or failure of the crossbar. Plus, if you re-rivet the bracket, it's likely that you will eventually run into the same corrosion issue in the future forcing you to relocate and drill even more holes in your crossbar. The thru-bolting provides a very secure connection for the bracket. I used #10 machine screws with a round head to avoid any sharp edges on the top of the crossbar. A flat washer helps to distribute the load across the rounded crossbar surface.
The other thing I did was to bed the dead eye bracket in epoxy where it contacts the crossbar. The epoxy acts as a galvanic barrier to help prevent corrosion from the stainless steel contacting the aluminum. I used West Six-10 epoxy which worked quite well.