If 4 C batteries, that's approx. 6v. I'd stay in that area if you're swapping to something else. But I recommend staying with the C batteries. They average 3800mah each. Someone else may have more knowledge than me, but I think you're going to be hard pressed to have better runtimes with lithium...
I had a Garmin Fenix 3, and if I'm not mistaken the data could be viewed from the watch, without a smartphone. But again, I could be mistaking that for something like barometric pressure or something.