Hi!

Straight to the point: how does one get a pet to have double energy regeneration and increased energy cap?

Exclude winning species events.

Exclude Tireless heart.

Before the Tireless Heart gene was introduced, and back before I joined BK, there was an event where you could get Festive Outfits that if used on a pet would change its skin and increase its energy cap by 5 (making it 15).

Items from the Warlock Exchange Station also can provide these benefits. Stimpacks allow a pet to regenerate energy in half the time for 3 days while Aromatic Beans increase a pet's maximum energy for 5-7 days.

Oh, that's it! The stimpack! I thought I saw that somewhere. Wait, Warlock Exchange Station? What is that?

It's been around during a couple different events with the latest one being the Egg Hunt that introduced the Hoppers. That's the only one I've been here for so I can't tell you any more about it than what I know from it. Users could find colored eggs around the islands while exploring and trade them in at the Warlock Exchange Station for cool items or pets. It was the top however many users who traded in the most eggs who were rewarded with the Hoppers. 

cool