How to design a continuously time-variant digital filter?
One approach would be to try to remove the frequency chirp from the observed data, thus translating all of the echoes to approximately baseband. I find this to be most straightforward by converting the observation to an analytic signal, then multiplying by a complex exponential whose instantaneous frequency is equal to $-1$ times the frequency chirp profile (while keeping its phase continuous). After dechirping the received data, you could then apply a lowpass filter to suppress any other sources that don't overlap in frequency with your chirp profile. If your followon analysis methods need to see the frequency ramp, you could reapply the chirp again by multiplying with another complex exponential.

The passband width of the lowpass filter defines how tightly around the transmitted tone that you reject other frequency components. The width of the passband would also need to be selected while taking the expected two-way propagation time of the transmitted signal; at time $t$, assuming a low-to-high frequency chirp, you might be transmitting frequency $f_t = f_c + \Delta f$, while the receiver is observing a delayed version of what you transmitted some time ago, for instance $f_r = f_c$. Your lowpass filter must have enough frequency coverage to cover the frequency slew of your chirp profile over the expected range of time delays. At the same time, though, you have an incentive to make the passband width as narrow as possible to reject other signal sources that are nearby your chirp profile in frequency, so as so often occurs in engineering, you have a tradeoff to examine.