As the manual shows, the Concorde's trim was conventional (and not by fuel transfer) and that the control surfaces shifted the neutral position as would be expected (here's a schematic).
Moving the fuel around is not managed by the pilots, rather the flight engineer. It's a long-term process that has to do mainly with the lift force shifting aft as the plane goes supersonic, and shifting forward again as it slows down. Resorting to an aerodynamic solution for the shifting center of lift was not acceptable:
To make the adjustment by aerodynamic methods as in the subsonic aircraft is not feasible because any deflection of flying control surfaces would have to be made throughout the supersonic cruise and would cause unacceptable drag (Concorde History).
Delta wings do not need a trimmable horizontal stabilizer (THS) by default since the elevons are far back from the CG and are able to control the pitch. There are few deltas with a tail, like the MiG-21, but that's for increased maneuverability since it's a fighter.
A list of 59 tailless deltas can be found here.
So, once the researchers and engineers concluded the delta wing is the best choice for supersonic flight in the Mach 2 regime, there was no need for a THS.
Hypothetically adding a tailplane
Concorde was meant to be economical (on paper at least). If the sizing remained the same -- the same number of passengers in the slender 2+2 seating -- then the addition of a tailplane would have reduced the main wing area (let's assume that offsets the weight of the added mechanical complexity). But the fuel capacity and thus range would have suffered. Now the Concorde is no longer a transoceanic plane.
If the range was fixed, then the payload would have suffered, considerably raising the fare (in 1980 a Concorde ticket cost £600 -- \$3,900 in today's money) and making the sales team job way harder than it already was (the sonic boom and smoky engines concerns).