My guess would be some form of gene editing, where his sperm wouldn't be rejected by the woman's immune system as foreign.
Gene editing likely, though the actual process would likely have little to do with immune system rejection. In general, a female's immune system already treats sperm and fertilized eggs of the same species as invaders/parasites to eliminate (though this also means both have ways of evading that for species to successfully reproduce with each other, which the dog sperm might need so you're not entirely off the mark, just not the main point of concern here). Reproductive isolation is mainly caused by two factors:
- The genetic compatibility: Species that engage in sexual reproduction normally get half the necessary chromosomes from each parent. Humans have 46 chromosomes in total, so sperm and eggs of humans would transfer 23 chromosomes each. Dogs have 78 chromosomes so they'd be getting 39 from each parent. This is often the most significant block of reproductive isolation because it's a very fundamental incompatibility between species. (There's possibly other genetic compatibility issues beyond that. But I'm not really familiar with all the nuances there.)
- The physical barriers: Assuming you can overcome the fundamental genetic incompatibilities, the environment the sperm cells live in and the environment egg cells fundamentally differ. They are very sensitive to temperature (hence the dangly external balls in many male mammals) and pH variations. Also the basic internal structure and composition of mucus inside of the female's uterus may be very hostile to the sperm trying to reach the egg, and a different species sperm isn't likely to be adapted to overcoming these environmental and physical differences.
- And finally, should you be able to overcome the genetic compatibility and push through the physical barriers, then you still need to worry about immune system responses. Which will almost definitely be partially baked into parts of the physical barriers and get triggered by certain stimuli.
Rex is a mutant? How could Rex be able to get a human girl pregnant?
The real answer is: Magic or Sci-fi technobabble BS. Because unless Adult King is holding out on some truly shocking research, we have very little idea how this could be possible. Barriers to such inter-species breeding are still far too significant to be explainable in any sort of reasonable terms, so in this case the less detailed the explanation, the better. Because no amount of explanation would really explain it. Thus "treatment with the side effect of making him able to make human's pregnant" is about as good an explanation as any.
