Could the chance of catching pets be increased just a little bit?

I keep getting the pet down to super low health and I still can't catch it.

Tried 4 times (that's 30k), is it supposed to take like 100k worth of traps to catch one?

The Trap system will be revamped as soon as I get a chance to do so. The new system will be much more reliant on your actions rather than just pure luck.

The overhaul will probably be a two part thing. Once initially and then further improvements when the exploration system is moved to the new layout.

In your example above, you just had bad luck. At that health value, it should have taken ~3 traps to capture.