I suspect this might not be possible.
In the first case (blah/?unit1={A}&unit2={B}) you're accessing a static page (.../blah) with two arguments.
In the second case you want to produce a new page dynamically using run-time specified values.
It sounds like (and correct me if I misunderstood) you'd like the page localhost/unit1={A}/unit2={B} to be available on demand, before any other request is made.
This would mean any values of A and B must be allowed - which isn't supported by the system.
You'd need something that detects requests for a URL, then creates something to serve that page without processing arguments (only the URL). As far as I know, this isn't possible with LabVIEW Web Services.