No, the TRE can't do it for an
initial JAR licence. The CAA must observe the ride along with the TRE. That's the drama.
See if the TRE can't do the organising for you, the TRE should have known about requiring the observation from the CAA. Also, there's the fee to sort out. Where do you plan to do the LST for the CL604?
If it's not in the UK, you'll be up for all the costs, ie transport and hotel. Expect to pay business class prices.