Synthetic might be a good choice here, splitters often don't get their oil changed by hours but by time. I mean, I burned wood for many years and I never changed it on hours, but by seasons, start and finish or whatever. So if you do neglect to change in in a timely manner, the Syn might last a bit longer, although not a huge difference, but still I'd feel more comfortable with syn. A 5w40 sounds excellent for year round use, and never have to worry about protection.
If you decide not to go syn, then a good 10w30 or 10w40 would work well too. I just don't see the need for 15w50, even though it's a great oil, just not in my book for this app. 15w40 would be good only IF pull starting is not an issue in cold weather. I wouldn't be afraid to use 15w40 below 32F, but it might cause a starting issue. I never split wood much below 25F, so that slight bit below the 32F recommendation for 15w40 wouldn't worry me especiall on an air cooled engine.