I haven't found recording software to be that bad, even in a world as huge as yours, it's singleplayer with no plugins and no intense redstone machines, so I can almost guarantee you'll be fine.
If you would rate your computer at a 2 or more on the "toaster to 10" scale, then you'll probably just drop 5 frames while recording. The game is easily playable at anything above 20.
As to software recommendations, I'll show what I use, it's free, and probably the best free recording software out there (with the maybe exception of shadowplay if you have nvidia GeForce gear). It's Riva Aftertuner, which comes attached to MSI afterburner, which is essentially a overclocking device, but it allows for some fun performance tweaks as well. Here's a vid on install/tutorial:
Common problems when downloading and how to fix: Check the box when it asks if you also want to download RivaAftertuner (that's what actually contains the recording software, and you can't get it separately). Also, for some reason, it makes you download two things. So your "installer" app or whatever should run twice. The first time it installs MSI, the second time it adds all the Rivatuner stuff, so don't close the second install window.
Also, only you would find a mesa biome, post screenshots, and say "I discovered another mesa biome; I haven't actually been to the surface yet" I can't think of many other people who would just stay and keep mining, rather than immediately digging to the surface :P.
Always fun to check out this stuff, you have a lot of interesting things to show and say.