You're kind of going two different directions there.
If you want to work with animals in a more clinical setting, DVM +/- an MPH or ACVPM is probably a good option. And then you also have the choice of: are you going to go into clinical practice, or go industry/government with a public health/epi focus? The issue is that like others have said, there's not really a market for an "infectious disease veterinarian" who would work at a specialty hospital and get called in for tough or unusual cases.
For a more research based/laboratory diagnostics focus you may want to consider a DVM/MS or DVM/PhD. Exactly what goes on depends on the size and nature of the facility. I'm a fairly experienced high containment researcher and have spent time at a few different facilities. Some are mostly just research, others also have a heavy focus on diagnostics and act as reference laboratories.
Another option for you would be to consider laboratory animal medicine. Despite being a veterinarian myself, since I'm on the research side, I work closely with our lab animal vets during studies (which include in containment too). Pretty much any research facility that has animals in any capacity will have a clinical veterinarian separate from the research side. For solving puzzles, maybe pathology would strike your fancy also.
If you're not in vet school yet, figure out if that's definitely your intention and worry about that first. Some experience and education will help you narrow the field.