I use a similar shop-made wooden mallet for disassembling pens.
I also use the metal transfer punches.
To minimize damage to the mallet, I use the end-grain section of the head as the point of impact.
Yeah, that's not where most would expect impact to occur. End grain is much tougher than edge grain so smacking a metal transfer punch doesn't make a huge mess.
This also puts 100% of the mallet's mass directly inline with the transfer punch and allows me to tap more gently.
It provides a wider and flatter target to hit the transfer punch with vs. making contact via the side of a cylinder when swinging a mallet by the handle.
The one caveat is I wouldn't do that with a Red Oak mallet. Red Oak likes to split and smacking its end grain might tempt it too much.
White Oak won't do that…it does NOT like to split. Elm would be great (best way to split it is with a chainsaw or explosives).
I'm using Ash currently and have had no problems. I've used Maple as well with fine results. Poplar is a bit soft and light…works but it's not pretty so at least you won't mind if it gets banged up.
Shop-made mallets are a good project for learning a new tool…try making one entirely with a skew (or whatever other tool you might not be entirely comfortable with).
The dimensions aren't overly critical. Handle diameter is whatever feels good to you. Handle length needs to feel good as well.
Any 2.5"x2.5" blank of 10-12" length is a fine starting point. As soon as it's roughed into a cylinder, the head is pretty much done (taper it if you'd like, sand it if you want)….all that remains is shaping the handle and parting it off.
Larger diameters work of course but tend to be heavier in the hand (depending on species used).