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Yeah, I think reclamping would work. It looks like shars has wide jaws for 6" vises up to 18 inches wide. For 4" vise, though looks like they only have 5" jaws. You could makes some as well.

You are machining out past the edge of the vise. Yes, it vibrates up and down even though it is 1" plate. For a part this wide you should use wide jaws, two vises, or clamp to the table.