Using RTG Tools:

rtg vcfsubset -i input.vcf.gz -o output.vcf.gz --keep-format GT,DP,GQ

I would go for using SNPSift program This is an instance of how I use it to extract: position, ref allele, alt allele, Allele Depth (AD from genotype field), and functional annotation by gene ID.

!/bin/bash
for i in *.vcf;
do java -jar /data/software/snpEff/snpEff/SnpSift.jar\
extractFields "$i"\
POS REF ALT GEN[*].AD ANN[*].GENEID > "filt_${i}";
done

Have you come across SnpSift in your search for tools to do this?

http://snpeff.sourceforge.net/SnpSift.html

Section 10 in the link above (SnpSift documentation) could potentially be useful: 10. SnpSift Extract Fields