OK, not sure how your company works it but...
Any sector where you are operating as the Pilot Flying, hence making the decisions and running the sector can 'normally' be counted as a PIC U/S sector. This does not need to be given by a TRE/TRI but just needs to be countersigned in your logbook by the Captain giving you the sector and therefore 'supervising' your performance.
To be assessed correctly for a LHS position you MUST have an ATPL before you can become a Captain! If the issue of the licence is to contain the type you are current on, as most do, then you only need to tell the TRE/TRI at your next 6 monthly sim that you wish to upgrade your licence. He should then tick and countersign the ATPL licence skills test box on the CAA form. The sim is run as normal however you must demonstrate a manual, raw data ILS to minimums.
With your B737 time I cannot belive you do not have the minimum required PIC U/S time. This IS counted as PIC time towards your upgrade.
As long as you can present the CAA with evidence of running the sector from the RHS and prove it with signatures from the company or the Captains you will have no problems.