Need more space
It looks like your hanging area is kind of out of the way. Have you thought about just abandoning the tree hanging and sinking a couple of poles? Most of us are stuck doing this in the middle of some tiny yard but you look to have some space to play with. As long as you have the room that's kind of the best of both worlds. You'll have some nice shade but don't have to worry about your straps or other hardware getting sap or other goo on them. A nice pair of 6x6 square or similar round pole should be just right.
I'm kind of in the same boat with a need for space. I have a pair of palm trees in my back yard that are just about the right distance apart to hang. I'd like to be able to hang at the same time as my kids though. I need a 3rd support but it will end up right in the middle of my yard. I'm kicking around the option of using a metal pole but cementing a sleeve in the ground so it's easy to remove. If I can't sink the sleeve deep enough to get sufficient strength I might be stuck with some of the other suggestions here (spikes, guy lines, line to the fence, etc). I'm a cheapskate so I'm trying to source the materials for cheap or free.