You can fix this without uninstalling, but you will need to edit the registry and copy around some files. If you aren't comfortable with this, then try shiverNZ's method above.
I purchased and installed Sins and the expansion at the same time and I get an error about fmod.dll being missing because Entrenchment was installed in c: root folder.
I copied my "c:\Entrenchment" folder to the sins install folder:
32-bit windows "C:\Program Files\Stardock Games\Sins Of A Solar Empire\" or
for 64bit "C:\Program Files (x86)\Stardock Games\Sins Of A Solar Empire\"
For the rest of the post I will be using the 64-bit path
You also need to copy a bunch of files from "c:\" into "C:\Program Files (x86)\Stardock Games\Sins of a Solar Empire\"
These files are ConvertData_Entrenchment.exe, Entrenchment - ReadMe.txt, Sins of a Solar Empire Entrenchment.exe, sinsentrench_firewallgdf.exe and possibly msdia80.dll. Note: UninstHelper.exe already exists in "C:\Program Files (x86)\Stardock Games\Sins of a Solar Empire\".
Next you need to edit the registry. I use regedit.exe (In Vista Click the windows start button and type regedit and enter).
For saftey you should backup the keys you are modifying (e.g. use menu File -> Export...)
Hit Ctrl+F to bring up the search dialog and search for Entrenchment.
Hit F3 to continue finding until you get to a key which looks like this (Mine is for Vista64):
"HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Uninstall\Sins of a Solar Empire - Entrenchment"
Modify the paths in the fields "DisplayIcon", "InstallLocation" and "UninstallString" from C:\<filename> to "C:\Program Files (x86)\Stardock Games\Sins of a Solar Empire\<filename>"
Also for key "HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Stardock\Drengin.net\sinsentrench"
Modify the Path field from "C:\" to "C:\Program Files\Stardock Games\Sins Of A Solar Empire\"
Next run Impluse and launch "Sins Of A Solar Empire - Entrenchment" - it should now run fine.
Lastly, delete the folder "C:\Entrenchment" and any of the entrenchment files in "C:\" that you copied earlier to the Recycle Bin (just in case).