Why are you using a raised cosine filter in the frequency domain? A matched filter is a system that convolutes the signal input signal with a time-inversed replicate of the transmitted signal, in the case of the radar, or the non-modulated transmitted signal in the case of a communications system. By knowing that a convolution in time is a multiplication in frequency you'll be able to perform the matched filter.

First of all, though you may know this, convolution in time domain is equivalent to multiplication in frequency domain, and vice versa. So, if you want to use a smoothing window in the frequency domain you would have to use multiplication. That being said, the window function that you are using probably creates a signal in the time domain (check that), so you either have convert it to frequency domain, or convert your signal in the frequency domain to time domain (and perform the convolution).

I understand what you mean, but still you could use the LFM chirp as a carrier instead of a normal cosine or sine.

What do you mean by full half moon? Certainly, the frequency would change. Nonetheless there are techniques to embed data into it a chirp signal, and still have good radar performance

It's a data modulated chirp. You create a data signal, and use it to modulated the LFM chirp. So you can use the signal for radar and communication