No aligner (at least the ones I know) generates a bam (alignment) file that will give the straightforward information especially one you are looking for above.
Normally, the tools that are used to call for long Indels and other SVs using a BAM file will also generate an output file containing information baout which reads were used to detect these Indels or SVs. Pindel, breakdancer etc. are some of the example.

If you are looking for reads spanning a 1 kb deletion, no aligner will allow this big gap for the alignment so that you can directly extract the reads from the bam file. The other option would be to look for reads that are soft clipped and if you find many reads that are aligned as soft clipped around a given region (breakpoint) then it can help. Other, strategy would be find the pair of discordant reads that have been aligned around the indel with insert size less than the actual one but these reads will not be spanning the Indels.