The 153C uses the weight of the train to operate. Therefore, it must react to the lightest piece of rolling stock, and its adjustment screw must be set at exactly that point. Also, it must be located at just the correct spot, between the main track circuit and the controlled track circuit. (The controlled section needs to have its center pins removed at both ends.) It must be set for the length(s) of the trains that you plan to use.
Taking all of the above into consideration, it's best to do some trial and error when setting up a controlled section with the 153C. If you plan always to use the same two trains, with the same speeds, and the same loads, you can make a very nice display layout. But if there are variables, (especially in regards to train lengths) then it becomes more problematic.
Electrically, it's very easy. The contactor is basically just a single-pole, double-throw switch. The "common" terminal gets track "hot" and the "normally-closed" contact gets wired to the center rail of the section of track that gets controlled (stopping the train within it.)
I am not familiar with the signal you wish to use, but assuming it will operate in the same voltage range as the trains, and assuming it has "common," "red," and "green" wires, you would just connect the common wire to transformer return (outside rail connection), and the green lamp wire to the same terminal on the contactor as the controlled track section. The red lamp wire would go to the third (normally open) terminal on the 153C. Without knowing exactly the make-up of the signal, I would say that you don't need to worry about using the transformer you have at hand, because the scheme uses just track "hot" and "return" to work.
When train #1 hits the contactor, it shuts off the controlled track section and also the green lamp. It turns on the red lamp, and train #2 loses power until train #1 clears the contactor.
A relay, in this case, would only offer slight advantages, that being (1) the inherent reliability of the insulated-rail/relay combination, (no weight issues) and (2) the ability to keep the signal voltage separate from the track voltage due to separate relay contact poles. But, if you, being both the President and the Signal Foreman, wish to use the historical method, who am I to recommend anything else?