Without knowing much about the curriculum for an MDiv, If getting the MDiv degree is important to you, I’d probably do that first. Depending on your veterinary job, it may be hard to find time (and money!) for MDiv coursework if you’re a full practicing veterinarian. Sure you could probably find a part time veterinary job (certainly in todays market you could), but that assumes you can get by on a part time salary and we don’t know what the future will look like both for you and for the profession. Demand may be different by the time you graduate. Sure you might have a job that leaves you plenty of time for courses in the evenings, but you might not. An MDiv could be a decent backup plan in case anything happened during vet school and you were unable to complete the program. I think as long as you can explain to programs why you want to be a vet instead of/in addition to a minister you’ll be fine.
With that said, I know many people who are deeply involved (both in paid full time positions and on a volunteer basis) in ministry without formal degrees. Obviously your religion/denomination may be different than mine, but to me, if you’re wanting your main career to be a veterinarian, you can do a lot of self study and online learning you’re interested in even without a formal degree. Pursuing a degree isn’t wrong by any means, but if I were in your shoes I would probably just do the DVM and seek out resources outside of a formal degree program for more in depth study as time allows you to do so.