narrow band frequency response creation and general FR question
Let's assume that speaker, free space and microphone can be modeled as linear systems and that the overall system transfer function is given by $H(\omega)$. Then the received signal is given by
$$
B(\omega) = A(\omega)H(\omega),
$$
where $A(\omega)$ is the signal sent to the speaker. So, yes, if you know the transfer function $H(\omega)$ you can calculate $B(\omega)$ and it should align with what you record with the microphone.

To determine $H(\omega)$ in practice you will have use discrete-time signals and play a unit impulse $a(n) = \delta(n)$, record the received signal $b(n)$ and calculate the transfer function with help of the discrete Fourier transform (DFT):
$$
B(k) = DFT[b(n)]
$$
Now the maximum frequency $\omega_m$ you are able to analyze depends on the sampling rate $\omega_s$ with which the samples of $b(t)$ are acquired after the microphone to obtain $b(n)$: $\omega_m = \omega_s/2$. Frequencies greater than $\omega_m$ can neither be measured nor calculated and will influence your result because of the Aliasing effect unless they're filtered with a low pass filter before sampling. An alternative approach is to transmit a signal $A(k)$ that only contains frequencies in the range you're interested. The transfer function for this limited frequency range can then be calculated by
$$
H(k) = \frac{B(k)}{A(k)}
$$
In this case frequencies lying outside this range won't affect your result as long as it's a linear system.

A comment on processing power: the sampling rate influences the clock rate in a real-time system and doesn't affect complexity otherwise. The cost of the DFT operation (implemented by the FFT algorithm) increases with the length $N$ of the recorded signal $b(n)$. The greater $N$ the greater the frequency resolution of $B(k)$.