With BEDOPS bedops and bedmap set and mean operations, and Kent utilities to build 1kb windows:

$ tr -d ',' < elements_with_commas.bed | sort-bed - > elements.bed
$ fetchChromSizes hg38 | grep -v '_*_' | awk -v FS="\t" -v OFS="\t" '{ print $1, "0", $2 }' | sort-bed - | bedops --chop 1000 - > hg38.1kb.bed
$ bedops --everything elements.bed | awk -v FS="\t" -v OFS="\t" '{ print $1, $2, $3, ".", $4 }' > union.bed
$ bedmap --echo --mean --delim '\t' hg38.1kb.bed union.bed > answer.bed

Replace hg38 with your reference genome name (e.g., mm10 for mouse, etc.).

You can also use process substitutions to avoid creating intermediate files, bundling the set and mean operations into a more efficient one-liner:

$ tr -d ',' < elements_with_commas.bed | sort-bed - > elements.bed
$ bedmap --echo --mean --delim '\t' <(fetchChromSizes hg38 | grep -v '_*_' | awk -v FS="\t" -v OFS="\t" '{ print $1, "0", $2 }' | sort-bed - | bedops --chop 1000 -) <(bedops --everything elements.bed | awk -v FS="\t" -v OFS="\t" '{ print $1, $2, $3, ".", $4 }') > answer.bed

However, the first set of commands is probably easier to read, troubleshoot, and modify.

Another demonstration (and related Biostars answer) here, showing use of the --sum operation, instead of --mean, though it is the same principle: How can I bin my bed files into 500bp bins?

The bedmap command offers several score summary operations, in addition to --sum and --mean. Median, min, max, weighted and trimmed means, etc. Run bedmap --help or review the documentation for a full listing.