Should I keep my 700R4 behind a 5.3 LS or switch to a 4L60E?

Both work. The 4L60 is the renamed 700R4 — the electronic 4L60E is essentially the same transmission with computer control.

Keeping the 700R4: most LS engines sit about 0.400 in shorter at the crank flange, so you need a dished (or dual-pattern) flexplate with the 700R4's 10.75 in converter pattern and a converter adapter sleeve. Kits exist for exactly this. It shifts hydraulically, so no transmission controller is needed.

Switching to a 4L60E: Speedway notes LS swaps almost always use the 4L60E so the computer can control shift timing and line pressure, which gives far more tuning flexibility. You'll need a computer that runs it (the factory PCM it came with, or a controller like Holley's Terminator X Max).

A reasonable rule: keep the 700R4 if it's healthy and you want simple; go 4L60E if you're buying a donor engine and transmission together anyway. More: do I need a spacer for a 700R4?