Aligning to the whole genome, as is sounds like you've done, is the right approach to take. This related question discusses whether it is useful to subset the genome for alignment; you want to avoid subsetting to prevent spurious alignments.

Specific to your question, your capture targets are not useful for speeding up the realignment step. Step 6 is finding potential regions of mis-alignment based on the full set of reads, which is a separate problem.

My suggestion for using your BED file would be to evaluate the effectiveness of Hybrid Selection. Picard has a commandline program for this: http://picard.sourceforge.net/command-line-overview.shtml#CalculateHsMetrics

which provides a number of useful metrics to assess how well the hybrid selection worked: http://picard.sourceforge.net/picard-metric-definitions.shtml#HsMetrics

After finishing SNP calling, you can also subset your calls to your regions of interest with BEDtools:

intersectBed -a snp_calls.vcf -b target_regions.bed -wa > filtered_snp_calls.vcf

for downstream analyses.