Okay, I've given this a go but I'm still not getting anywhere. I'm sure it's me but here's my set up in a little more detail:
* Limited user account: Fred
* Privileged user account: Barney
Logged in to Windows as Fred, open ConEmu window as Fred.
Batch file in the ConEmu program directory contains the following (as per your guidance):
admin-psh.cmd
powershell.exe -new_console:a
ConEmu tab configured as:admin-psh.cmd -new_console:bnu:"<Barney>:<BarneysPassword>"d:"C:\Users\barney"
Now it seems to work - I click the link, I'm prompted to elevate and a PowerShell tab opens at 'C:\Users\barney' - however if I do a 'whoami' it politely tells me that I'm Fred.
Am I still missing something? :)