Would the tweeter resonance frequency be visible from full speaker sweep frequency response measurement, with DSP crossover in place ?
You might see a bump in the response, but it becomes most evident when there is another impedance (a simple resistor for example) in series with the tweeter. When a passive, series crossover is in place (your capacitor), the signal voltage is divided between series elements in the same proportion as the impedances. For example, if the impedance of the tweeter at a given frequency is 5 Ohms and if the impedance of a resistor in series with the tweeter is 2.5 Ohms, then since the impedance of the tweeter is 2/3 of the total series impedance (and twice the impedance of the resistor), 2/3 of the signal voltage will appear across the tweeter (and the voltage across the tweeter will be twice greater than the voltage across the resistor). The impedance of any driver increases sharply at the driver resonance. (Note that this is consistent with the fact that very little current is needed to excite the driver at the resonance point and that a small amount of current at a given voltage implies high impedance.) Since the impedance of tweeter will most likely increase sharply at the resonance point, the effect is that the proportion of signal voltage that appears across the tweeter at this particular frequency is a greater proportion than at frequencies both above and below the resonance. In other words the crossover doesn't do what it is supposed to do because it is designed with an assumption about the tweeter's impedance (the value you entered into the calculator to obtain the capacitor value), and this assumption isn't valid at the resonance.
If the frequency space between the resonance and the crossover point is adequately great, the impedance of the crossover element in series with the tweeter will be adequately high - it increases with decreasing frequency - such that the tweeter's impedance will be small (in comparison to the series filter's impedance) in spite of the fact that it increases sharply at that point. Many speaker builders nevertheless prefer to use a series notch filter even if the resonance point is two or more octaves below the crossover point. A series notch filter consists generally of a resistor, capacitor and inductor in series, connected in parallel with the tweeter, i.e., a shunt across the tweeter. It effectively lowers the impedance of the tweeter at the resonance, because it is sort of like a short circuit bypassing the tweeter but present only at the resonance frequency and not a complete short. This cancels the effect of the natural increase in the tweeter's impedance at the resonance, thereby allowing the high-pass filter in series with the tweeter to do what it is intended to do. (If there were no impedance in series with the tweeter, the notch filter would not have any voltage-lowering effect on the tweeter. In this case the effect would only be to increase the total amount of current at the resonance point.)
There are online calculators for determining the value of the resistor, the capacitor, and the inductor. The order in which you connect them does not matter, all that matters is that they are in series and connected across the tweeter terminals. The capacitor blocks frequencies below the resonance point, the inductor blocks frequencies above the resonance point, and the resistor establishes a floor of sorts for how low the shunt impedance will be at the resonance point. Trial and error may be needed to get the values optimized, especially if you don't know the actual impedance of the tweeter at its resonance. Clearly it is better to minimize the potential problem by insuring that the crossover point is at least two octaves above the resonance point, but this depends on how steep the crossover is. This is one way in which a steeper filter is better, because the steeper the high-pass filter is, the greater its impedance will be at the tweeter's resonance.
Since you will play the sweep tones with the simple high-pass filter in place, in series with the tweeter, you might be able to hear the resonance point, or see it in the recorded measurements. And if you decide that it will affect the sound quality, it may be possible to deal with using the DSP crossover you are using, depending on whether it will permit you to define a parametric EQ notch in addition to the high-pass filter. But if the DSP isn't able to do this for you, then depending on whether you think it is something that needs to be dealt with, you may decide that it is warranted to place a series notch filter in parallel with the tweeter, across the tweeter terminals.