I don't know of a preexisting tool that can change the coordinate origin at the DICOM level. Our DICOM->DICOM workflow is more of a bundling of separate tools, so if you can fix it in nifti space, you can run ADIR_nii2dicom (also in the .zip) on the result (with the original DICOM as an input), and that's the same thing the dicom->dicom workflow would have done anyway. At the nifti level, we use in-house tools to correct origins, but I know it can be done graphically with fsleyes and I think fsl has a command-line tool also. From your image, generally I wouldn't consider that origin to be "bad". mri_reface has built-in detection and correction for "bad" origins, but that typically means origins outside the FOV. This is close enough that I wouldn't expect the registration to go bad from that alone (and it's not bad enough to trigger our internal correction), but adjusting it still might help. If you correct the origin at the nii level, the new DICOM post-conversion will inherit the geometry from the adjusted .nii and produce adjusted DICOM.

Another option is to run your own registration to MCALT_FaceTemplate_T2.nii with external tools and use the -regFile option. 

If you have a 3D T1 or 3D FLAIR to accompany your TSE, you might also run both at once using the multiple-inputs function, with the 3D image first. When run that way, the 2D image is aligned to the 3D image, and the 3D image defines the alignment and warp to the template. Typically that solves issues with these 2D MRIs and PETs that have smaller FOVs in Z, because the intra-subject reg is more robust than subject-to-template. I think you would also need to use .nii files for this solution, but again you can convert to DICOM with ADIR_nii2dicom after the fact. 

Chris