Pulse shaping with root-raised cosine filter for a non-Gaussian channel I am here to check whether my line of thinking is correct.

Root-raised cosine (RRC) filtering is one way to minimize intersymbol interference (ISI). Hence, we apply matched RRC filters on both sides of the communication link.
We can adjust the roll-off factor $\beta \in \left(0,1\right]$ to trade-off the bandwidth (BW) for a better signal quality at the receiver. The borderline cases of $\beta$ are $\beta=0.01$ and $\beta=1$, where we &quot;sacrifice&quot; 0.01 BW and 0.5 BW, respectively.
Now, this is usually when the impulse response function of a channel is assumed to have a normal distribution (or similar-ish). In other words, the noise is linear (i.e., AWGN)
If this is no longer the case and the impulse response of the channel does not follow normal distribution, would RRC still be a viable solution? - I guess not but please correct me if I am wrong.
There are many related questions on this site; make sure to review them. To summarize:

Correct.
The roll-off factor $\beta \in [0,1]$. If $R_p$ is the pulse (or symbol) rate, then the pulse bandwidth is $B=(1+\beta)R_p/2$. When $\beta=0$, the pulse is a sinc and the bandwidth is the smallest possible. For $\beta&gt;0$, the factor $1+\beta$ is the &quot;excess&quot; (not &quot;sacrificial&quot;) bandwidth.
The AWGN channel does not distort the input signal, only white Gaussian noise is added to it. The density of the noise is Gaussian (normal), not the channels' impulse response. The noise, being a signal, cannot be linear or non-linear (only systems have this property).
RRC and other Nyquist pulses can avoid ISI in many different channels. For example, in  a frequency selective channel, after linear equalization the received signal has no ISI. RRC pulses remain the most popular in practice, in all kinds of channels.