How do I know whether AR1 noise is an appropriate null hypothesis to test against?

It is usually an appropriate null hypothesis if the theoretical AR1 spectrum is degreesa good model degrees for the power decay in the observed spectrum. I recommend to simply visually compare the two power spectra:

X=rednoise(200,.8);
[P,freq]=pburg(zscore(X),7,[],1);
aa=ar1(X);
Ptheoretical=(1-aa.^2)./(abs(1-aa.*exp(-2*pi*i*freq))).^2;
semilogy(freq,P/sum(P),freq,Ptheoretical/sum(Ptheoretical),'r');
legend('observed',sprintf('Theoretical AR1=%.2f',aa),'location','best')