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"Also, there is one metal line going from the mast to the aft area. I guess this hooks into the boom?"
That sounds like a topping lift. If you're not sure what the rigging correctly is--try to find other sunbird owners and make sure it is right.
Generally you don't want to mix metals and the question would be whether the originals are breaking because ? old age? Does it take 20 years for them to fail? Do other owners say they are a design error? Or maybe, someone just didn't set them properly.
There are all sorts of "pop" rivets and riveting tools, I'd suggest looking over some and contacting the manufacturers to make sure you are using one with the necessary quality. The chandleries, Grainger, McMaster, Jamestown Distributors, all should have a couple for you to choose from. It could be the pop rivets that were used were simply cheap, and moving up one size would give you the extra strength you're looking for. Or, a different brand may be all you need.