Ferris,
I am afraid that you don't (should not) have a choice, once an aircraft declares that it is following a TCAS RA, it is advised that you DO NOT give any contra instructions.
I suggest (as you said) that you give Essential Traffic Information (but) to the other (conflicting) aircraft.
An important thing to remember, is that as ATC's, we get a reduced re-fresh rate (sweep update) on the SSR, that is to say relative to the aircraft equipment, and it may well transpire that what you and I see on the Mode C readout is out of date by 5 seconds, and could imply that aircraft have already crossed levels and cleared each other.
Zagreb is a classic example of 2 aircraft that had actually cleared each other, but due to the ATC instruction, the climbing aircraft actually descended back into the collision path - A tragedy that you and I would rather avoid I am sure.
As difficult as it may sound, if the situation has reached the point that aircraft are saying "GFA231 IS TCAS DECENT !!" The best we can do is close our eyes and say a silent prayer.
Cya