CGS 29-29 says the local issuing authority has '5 days' to get your prints out to SPBI. When SPBI completes your background checks and returns them to the PD (should take less than 72 hours), the local issuing authority should take no more than '1 week' to issue or deny your permit.
If 2-3 weeks or more have elapsed, you should be contacting 'records' at the DESPP and requesting when your background checks were received and sent back. If these time limits have been exceeded, you should file an appeal.
8 weeks is a part of CGS 29-28, but the concept of PDs having 8 weeks to process a permit application is the same kind of lore as 'Connecticut is a concealed carry state' and some of us are trying to correct that misinformation.
And no, you will not be dealing 'directly with the CSP'. You will be dealing with your local issuing authority to get the temp permit and then with the DESPP to get your permanent permit.