Finally came across a solution.
After weeks trying to figure out what I were missing, I've found that on my development machine I have a different Framework version (in details, I have v4.0.30319.18444 vs v4.0.30319.17929 on test machine). I still don't understand why I shouldn't be able to debug my application without having the exact Framework version, but the solution was to copy mscordacwks.dll and mscordbi.dll (both of them found in C:\Windows\Microsoft.NET\Framework) from the Test machine to the folder where the minidump is.
Reference: Managed Minidump Debugging in SP1