I was able to solve this myself. To build my selected values segments I was using variable with this expression for each filter field
If(GetSelectedCount($1)=0,Null(),'select/' & '$1/' & Concat(distinct $1,';') & '/')
When a field had spaces in its name, I had to enclose it in brackets when passing it into the variable or else the variable would not resolve:
But in doing so, the string "[Field 1]" would be placed in the app integration URL, which is apparently, bad syntax.
I changed the variable's formula to this, and everything started working:
If(Len(GetCurrentSelections()) = 0, Null(), 'select/' & GetCurrentSelections('/select/', '/', ';') & '/')
GetCurrentSelections returns a string of all current selections in a format where you can find various delimiters. Check it out in the help file. It bypasses the issue I was having with using the square brackets to get selections for field names like "Field 1". Also, it simplifies your implementation, because you only have to reference the variable once for the entire URL rather than one time for each field that may have selections.