I got it working, after doing two things:
Using the .NET Framework 2.0 version of the CLR Profiler - there's a separate .NET Framework 1.1 version, which is the one I had found initially. It probably doesn't work with CLR 2.0 and up; and
Running the CLR Profiler and application under an administrator account.
So, just to confirm, yes, the CLR Profiler (at least the Framework 2.0 version) does work for native applications hosting the CLR (it only inspects managed code, naturally.)