Hi Skyte, welcome to lombok lovers. Have you contacted the international school in Lombok, Nusa Alam? They teach in both Indonesian and English. I know that they do contract teachers every year, although I think maybe only qualified teachers - so not just to teach English. Still, it might be a good place to start to get to know the academic community.
Having lived in Lombok with my Sasak husband, I can tell you that finding work there is going to be pretty tough. Most expats I know who work didn't go there and find a job - they were there because their job sent them there. Otherwise they had started their own business which was successful enough to support them. OR they live outside of Lombok for half the year to work and make money, and then live in Lombok without the hassle of working for the remainder of the year. Some work online, and a few lucky individuals who have enough money and investments not to need to work. I also met some working for NGOs, but again, they were mostly recruited outside of Indonesia.
If you have spent time there you know that a good percentage of the local population are unemployed or barely work enough to survive. The government, rightly so, makes it difficult for anyone other than an Indonesian to work in Indonesia. There are plenty of Indonesians capable of managing a cafe or restaurant, or running a hotel - there is no need to get foreigners to do that, and so you will rarely see anyone other than an owner doing that sort of work in Indonesia - and even then, not often as the visa situation is tricky.
My advice to you is to work long enough in Perth to save enough money to last you 6-8 months - go and don't stress too much about finding work, but keep your ear to the ground for opportunities. When the money runs out, hop on a plane and go back to Perth and make some more (thanks to Air Asia that is pretty cheap to do these days). I've lived in Perth for many years, and worked in hospitality for quite some of that in the past. Years ago when I lived on Rottnest island, quite a few of the staff at the hotel and the lodge worked the summer there and spent the rest of the year in Lombok and Indonesia surfing and having fun. Hospitality is good like that - not great money but flexible and seasonal. I have to admit that was a pretty good life - best of both worlds really.
After a couple of years of doing that, you will know more about everything, and will likely have found some way to live the life you want.