I don't think so because the pDEST15 plasmid contains the ccdB gene. Expression of the CcdB protein kills E. coli by interfering with an important enzyme for DNA replication in E. coli (DNA Gyrase A) which leads to DNA breaks. In a lot of Gateway vectors, ccdB is included in the plasmid coding sequence between the att sites to reduce the number of transformants without an insert. The company replicates these plasmids in special strains of E. coli that either overexpress the ccdA gene in trans (the CcdA protein binds to and inactivates CcdB) or which have special mutations in the gyrA gene which prevent CcdB from binding to DNA Gyrase A. Once the unmodified plasmid is transformed into a normal strain of E. coli, it causes cell death.

You would need to have a strain that had both T7 RNA polymerase (for expression from the T7 promoter) AND was resistant to CcdB. I could be wrong, but I do not think such a strain is readily available and easy to acquire. Even then, with how pDEST15 is arranged once you induced GST expression, CcdB production would likely increase greatly as well, which might kill even a normally CcdB resistant strain of E. coli.

I think the easiest thing would be to either use another vector or to modify pDEST15 to remove the ccdB gene, and preferably add a stop codon right after the GST tag coding sequence so you don't have to worry about readthrough into some random vector sequence causing issues with protein solubility or something.