You could perhaps take the multiset union of your ChIP-seq datasets:
$ bedops -u chip1.bed chip2.bed ... chipN.bed > chips.bed
Then use chips.bed as the map file in bedmap, applying the --echo-map-score operator to retrieve score (value) data from the unioned dataset:
$ bedmap --echo --echo-map-score --delim '\t' reference_regions.bed chips.bed > reference_regions_with_scores_from_overlapping_chip_peaks.bed
If this does what you want, then you can set up a pipeline to do both steps in one line of code:
$ bedops -u chip1.bed chip2.bed ... chipN.bed | bedmap --echo --echo-map-score --delim '\t' reference_regions.bed - > reference_regions_with_scores_from_overlapping_chip_peaks.bed
This would run faster, since you wouldn't create the intermediate file chips.bed.
I may not fully understand your question, but hopefully this suggests options. Please let us know if this does not help.