Any tricks to help me transplant this without puncturing myself or damaging the plant? There doesn't seem to be a smooth spot to get a grip.
Use a towel or rag or newspaper cut in strips. Holding an end in each hand making a "u" around the base of the plant. Cross the ends over (sorta like 1/2 of a figure 8) around the plant and twist them to make a stiff "pot handle".
I use this method when transplanting my thorny 'pets' and it works very well.
I found a link that sort of shows you the general idea.
Here is a link that might be useful: transplanting cactus
With Crown of Thorns I put on my leather gloves when working with them. Any damage that may occur to the thorns is minimal. When moving cactus I do as Wanda suggests, usually using an old T-shirt cut into strips.
BBQ tongs if you need to reach in very far ...diana