As best as I can tell from your photos, those appear to be a type of Pop-rivet. You need to drill them out or use some sort of grinding tool to grind the heads off. Depending on what type and thickness of material is backing your new bumper, you may be able to use sheet-metal screws to install the old brackets to the new bumper, otherwise you'll need a rivet tool. Whatever tool and fastener you wind up using, be sure it's not too long so it doesn't penetrate the outer bumper skin.

Happy Motoring, Mark

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DrDKW