Is there a limit to how many times I can attack the same battlefield? If I send in a Skuttle, after it dies, can I send in other units to start a new showdown?

Yes, there's no hard limit. You can initiate as many showdowns at the same battlefield in one turn as you can afford to move units there.

Here's how it plays out with Scuttle Crab:

1. You move Scuttle Crab to a battlefield you don't control → it becomes Contested, a showdown starts.
2. During that showdown, Scuttle Crab dies (it's 0 Might, so any damage kills it).
3. When the showdown closes, you have no units there → the opponent keeps (or re-establishes) control.
4. Back in Main Phase / Open State, you move another unit in → battlefield becomes Contested again → a new showdown begins.

Repeat as many times as you have ready units and movement phases to spare. Each one is a fresh showdown. The only real limits are movement resources (units cost exhaust to move), timing (Standard Moves only during your Main Phase in Open State, not during an ongoing showdown), and the once-per-battlefield scoring cap (you won't get a second Conquer/Hold point that turn — Rule 465).

As FAQ #6149 states: "There can be as many showdowns in a turn as a player can legally enact, limited primarily by movement resources and ready units."