There are multiple Ruger .44 mag carbines in my family and they have all dropped deer out to about 100 yards, which is a long shot where we hunt. Both are great with hollow points around 220/240 gr.
One thing to consider, not sure on the rules there, but a .44 is considered a handgun cartridge in some states, such as MA (where my Dad lives). He had to get a pistol permit to buy .44 mag cartriges for his rifle (figure that logic out).
I have zero direct experience with .243 but have a friend who loves it.
Another "low kick" to consider is 7mm-08, recently got one for my wife, she loves shooting it. Bit harder to find bullets for if you don't handload, but it's common enough that basspro, cabella's etc, all have them, but Wal-Mart doesn't.