Basics:
You don't generally use NTC when bypassing it with relay. You can do it, but it's not optimal.
The idea of using NTC is that it's self-contained, there's nothing else. It's poor, but it's simple and cheap. It's lossy all the time, and you have the tradeoff between losses and inrush current, and you have quite a bit of inrush anyway, so it only works when it's nearly not a problem anyway. It also has problem with repeated on/off cycles during about 10-20 seconds, but you rely on it being nearly unnecessary anyway.
When you have a bypass relay, then you use a fusible resistor, or a standard power resistor with properly dimensioned time-delay fuse in series (with the resistor only; the fuse is bypassed too), or a PTC, or a PTC thermally coupled with power resistor in series. The idea is that if your precharge drive side fails for any reason, the fusible resistor blows; or the fuse with standard resistor blows before the resistor overheats. PTC needs to have enough cold resistance so it works like the resistor, but almost stops conducting (and hence, precharging) if the driver is stuck.
Driving based on time is waste of design effort. Measure the rectified output voltage (the one with the large caps) with a simple comparator, for example, and drive the bypass relay when the precharge is finished, i.e., the voltage is near the operating voltage.
If you have an MCU there anyway, it's even better to use it to measure the voltage rise and drive the bypass relay. The MCU needs to be powered up first through a secondary supply. This way, you can implement timeout too, so that you rely on the fusible resistor (or PTC action) only in a severe case of MCU malfunction.