1.- MATLAB already has command snr to directly calculate Signal-to-Noise ratios
command snr details available here
2.- the signal may have also have intermodulation, either from transmitter or caused by reflections along the channel(s)
The other 2 similar measurements to snr but not the same are Total Harmonic Distortion (THD) and Signal to Noise and Distortion Ratio SINAD.
Both also have dedicated commands:
Details about command thd available here.
Details about command sinad available here.
3.- Some literature also use Signal-to-Interference SIR along with SNR or in some scenarios where intereference >> noise, like some urban mobile communications, completely ignore SNR and use SIR only.
There's no command for SIR because deciding what is noise and what is interference is not a straight away task for a MATLAB base command.
4.- To calculate detectability first you have to define an acceptable probabilty of detection, or equivalently, and acceptable probability of error by error comprising both missing pulses and taking noise/interference as valid pulses.
Do you have this figure?
If you supply a sample of the signal I may be able to supply some MATLAB code for you to start with.
5.- To truly calculate signal power out of signal amplitude samples one has to include imedance.
Because you mentioned pulse radar you have to be careful with the time window or cycle used to calculate signalsignal power.
If you more than one pulse in same cycle used to calculate signal power you have to average, but then SNR is going to be averaged, meaning one cannot tell at what time within each cycle did SNR spike.