Convert bam file to highly compressible bam
You can just set the fields you don't need to *:

samtools view -h foo.bam | awk 'BEGIN{FS="\t"; OFS="\t"}{if($1~/^@/) {print $0} else {print "*", $2, $3, $4, $5, $6, $7, $8, $9, "*", "*"}}' | samtools view -bo smaller.bam -


This will set each read's name to *, but you can still see where its mate maps with the PNEXT and RNEXT fields.

The resulting BAM file looks like: 

*   83  chr1    3003308 37  51M =   3003157 -202    *   *
*   99  chr1    3003364 33  51M =   3003547 234 *   *
*   99  chr1    3003410 37  50M =   3003636 269 *   *
*   99  chr1    3003438 255 50M =   3003760 373 *   *