You might try the definition, like e.g.:
testFunction[x_, problemID_: 1] := (whatEverFunctionBody[x, problemID]) /; MemberQ[ {1, 2, 3, 4}, problemID]
That is one way of doing it, but that's not an option ;)
This is kind of tricky for options what generally happens is that the function throws a message when the options are not correct (out of bound). See for example how it is done in the built-in Plot function, you can see how the arguments are parsed et cetera:
Needs["GeneralUtilities`"] Plot[Sin[x], {x, 0, 1}]; PrintDefinitions@System`ProtoPlotDump`iPlotLayer1 PrintDefinitions@System`ProtoPlotDump`iPlot
