Is it possible, and is it statistically sound, to use diffHiC for differential HiC analysis when no biological replicates are available? I know edgeR lists a few options for calculating dispersion with biological replicates, but none of them seemed to good options. We can calculate dispersion for no bio reps with DESeq as well. But I'm less certain about whether it is possible (or statistically sound) to create the "fit" object from non-replicate data which diffHiC relies upon for the rest of the analysis (Ch. 6 onward). Has anyone tried this or know whether this is possible?

It's as possible and as sound as RNA-seq data analysis without replicates - that is, not very. All of the comments in the edgeR user's guide pertaining to lack of replicates are similarly applicable to diffHic.