Another way to keep using chi-squared but to avoid binning up the data is to modify the statistic so it works better for small numbers of counts/bin. There are several solutions. All these are purely empirical.

Gehrels (1986) suggested replacing the sigma of sqrt(N) with (1 + sqrt(N + 0.75)).

Churazov et al. (1996) suggested keeping sqrt(N) but calculating the N as an average over surrounding bins.

Mighell (1999, 2000) has two statistics based on estimators for the weighted mean of data drawn from a Poisson distribution.