I assume your "regions.bed" file is not huge otherwise you would end up with tons of files. You could go through regions.bed line by line and output the result. Here's an example using bedtools, but the same idea should be doable in bedops:

n=1
while read line
do
echo "$line" | intersectBed -a main.bed -b - -wa > region.${n}.bed
n=$((n+1))
done < regions.bed

(It's a bit unusual what you are trying to do, it might be that there are better ways of structuring your data)

Edit: Maybe this is better then having one file per interval:

intersectBed -a main.bed -b regions.bed -wa -wb \
| awk -v OFS='\t' '{print $1, $2, $3, $4, $5"_"$6"_"$7}'

chr11    13302    13303    1    chr11_13202_14981
chr11    13327    13328    1    chr11_13202_14981
chr11    13980    13981    1    chr11_13202_14981
chr11    13980    13981    1    chr11_13980_41480
chr11    30923    30924    1    chr11_13980_41480
chr11    30923    30924    1    chr11_30923_51480
chr11    51476    51477    1    chr11_30923_51480
chr11    51479    51480    1    chr11_30923_51480

Now the 5th column is an identifier of each interval in regions.bed.

You could use the following script as a general solution. You could either use BEDOPS bedmap --echo-map --multi-delim '\n' or bedops -e 1, depending on what detail of data you need. In bedmap, you could pipe in each region/line as a reference element via stdin, using main for map elements. In bedops you could use the main for reference and each region element as a map, each region/line piped in via stdin.

You could pipe the results of BEDOPS bedmap to split:

$ bedmap --echo --echo-map regions.bed main.bed | split -l 1 -a 5 - bedmap_result_

For each line in regions.bed, you would get files called bedmap_result_aaaaa, bedmap_result_aaaab, and so on.

See man split for more information on the -l and -a options used in this example.