What would happen to the turn-on transient if I add a capacitor in parallel to a transformer?
Transformer inrush typically has little to do with the load on the secondary. So the capacitor would have to go on the primary to have any effect. On the primary it will undoubtedly increase inrush current, though probably for a shorter period than the transformer. That's because at the point the device is turned on there will be an instantaneous voltage applied to the capacitor, making for a dramatic rate of change from zero. Capacitor impedance falls as rate of voltage change increases, so it will be close to zero in this case. The only thing limiting the current then is the series resistance.

The best way to reduce transformer inrush (which is caused by the delay establishing a corresponding magnetic field) is with an inrush current limiter. The three basic inrush current limiters are, in rough order of complexity: a fixed resistor with a time-delayed relay across it to short it out after the inrush period; a NTC thermistor; which automatically falls in resistance over time due to heat; and an electronic soft-starter.