Does the 1M solution which the metal strip rests in matter as long as the cation is the same metal as the strip?
No, but yes.
If you can guarantee the activity of the free metal is 1 - doesn't matter what the other components are (at least long as they don't react separately).
Trick is, many anions have quite strong complexing properties, so some of the cation will be complexed and you won't know its concentration.
Also, ionic strength of the solution will depend on the charge of the counterion, so it gets difficult to make sure the activity of the cation is exactly 1.
Sadly, plenty of fine prints which make the situation complicated and difficult to control.
A lot depends on how accurate your measurements need to be though, some of the effects can be ignored, some can be estimated.