Could you make alum with metals like beryllium, magnesium, calcium, tin, or lead?

Answer 1

No. "Alum" is a sulfate of aluminum and a monovalent cation like potassium or sodium. Similar compounds can be made if Cr or Fe is involved instead of Al, but these are all trivalent and the double sulfate still requires a monovalent second metal.

The metals you've named are all divalent (Be, Mg, Ca) or divalent/tetravalent (Sn, Pb).

Answer 2

Instead of sodium or potassium you can use one of the other group 1 metals like rubidium, caesium or, if you can find some, francium.