Some (all ? most ?) LABCs have different charges for minor electrical works. Mine charges about £150 or £225 - the difference being whether you can supply acceptable test results. So you may find that if you are qualified, and provide the right results (the results from an EICR should be fine) that the LABC charge is reduced.
But to clarify, there are only two routes to legally installing a new CU.
1) The person doing the work must be a member of an approved scheme and self notifies through their scheme.
2) The person commissioning the work notifies LABC before work starts, pays the fee, notifies LABC at appropriate points (for a CU this would be when the work is done), LABC inspects the work, and LABC issues a completion certificate. If the applicant can supply test results, then LABC may skip the inspection part, and may reduce the charge payable.
There is no other route
It is a common question, and no, there is no route for an electrician to "sign off" someone else work. The best they can do is test and provide results so as to reduce the need for LABC to inspect & test.