You can easily do this with a backpack and a roller bag. You are, IMO, missing out on a big part of safari. That is getting your rifle and yourself ready for the epic adventure. I understand the idea but from my experience it is worth the trouble to take your gun and a checked bag. I’ve been all over the world with my rifle and sure there are concerns along the way but having all my gear is worth it! Get a travel agent, your gun, and go on safari!
For me part of the process is using my own rifles with my own ammunition. You are travelling around the world extra few minutes dealing with checked luggage is nothing compared to the total travel time.