Dan, handling is always a subjective area but in my experience there is very little difference between the TTRS (sorry no experience in the S) and the RS3. Yes the RS3 is agile, flows nicely through fast sweeping corners and gives huge confidence if the corner tightens. It's still typical Audi in approach, get the entry speed right so the front hooks up then on the gas early and fire through. The RS3 exit speed is its party piece, with so much grip to play with you can use all that power and it's as if you've lit the afterburners as it rockets away. With the ESP 'off' you can get a couple of degrees of movement at the rear, just enough to settle the car and angle it through the apex.
If the TT has an advantage I'd say it might be just a fraction more keen to turn in initially.
All subjective as I say because the RS3 does everything I ask with no fuss or drama. It doesn't have that on tip toes feel of an Evo, which would move around a lot more at the rear through the same corners described above - that bit of oversteer was great fun on track but could be a little unnerving in the wet. Nor does it have the ultimate finesse of something like the R26.R in which you can feel exactly how much grip the tyres have left and place the car on the apex time after time. As an all rounder though it's much better than both of course.
Hope that helps you make your choice.