If the first few threads of a tapped hole are all messed up, the ideal tap to use would be a taper tap. Unfortunately, they're called different names. This type of tap would not have fully formed threads at the end. That makes it easier to start in an untapped hole, or in this case a tapped hole with messed up threads.
A bottoming tap would be the worst to use in this instance. This type of tap has full threads right to the end. Starting it in messed up threads will be difficult. It's important to remember that you don't want to cut a new set of threads, you want to clean up what's there. It will be very difficult to do this with a hole with messed up threads and a bottoming tap.
Here's an illustration that I hope will help. It's a little hard to read though.
Using the nomenclature in the illustration, a Taper tap would be ideal. It has a nice long lead into full threads. Easier to get it started straight and true and pick up the existing threads. The Plug tap is second best. The bottoming tap the worst. My guess is that a bottoming tap would pretty much guarantee failure in this instance.
If you cut a second set of threads, you will never get the new plug to seal. Cleaning up the existing set of threads is of the utmost importance.
Remember too, that it's a pipe thread. It's possible to tap it too deep to where the plug will not bottom out and seal the threads. You want to clean up the existing threads, but by no means do you want to go two or three turns deeper which will just make the hole larger. It's tapered, the deeper you go, the bigger the hole gets. I hope that makes sense.
If it's truly a pipe thread and IMO it pretty much has to be, your choice of taps will be limited. A pipe tap by it's very nature is tapered. This will be very helpful in getting it started.
Some background on me. I was a Die Maker for 30 years and drilled and tapped all the time. Many, many times on a daily basis.
As a side note, there is such a thing as a straight pipe thread. I've heard of it, but have never seen such an animal. I have no idea how a straight thread would seal. I believe that it doesn't rely on the threads for sealing there is a sealing surface provided that does the actual sealing. No sense going into all that here. That plug you provided the pic of has no other means of sealing and must be a pipe thread. That's been the standard for manual transmission fill and drain plugs for ages. It's inexpensive and effective.