The only way that I can think of doing this is by what looks like you've already done--create a parameter with a list of the desired measure names. The issue with this as I'm sure you've discovered is that you can only select one measure from a parameter control at a time--this might be a good one for the Ideas section, if it hasn't already been added!
Hey Eric -
There's not a great way to approach this based on the fact you want to lean on "Measure Names" - we can't duplicate the Measure Names dimension and/or easily reference it in an expression, which are the building blocks one normally uses to attack this sort of scenario.
Hopefully someone else can come up with a more original approach, but barring better ideas you could base this single report on a copy of the original data source....In the duplicate data source, hide all the measures you don't want people to play with. Yes, I know - lame, but it'll work in a pinch.
Right click on "Measure Values" (step 1), select "Show Quick Filter" (step 2), highlight all the measures you want to remove using the CTRL key and arrows, right click and select "Hide". Your Measure filter will now only show the unhidden measures (step 3). You can still view the hidden measures by clicking on the down arrow by "Dimensions" and selecting "Show Hidden Fields". All the hidden field will be grayed out. Hope this helps!