I’ve only had brown turkey and Hardy Chicago. They never branched much. They like to make multiple trunks. I don’t think trimming it back will make many branches unless you cut it back almost to the ground.
Is your tree trying to make more trunks? If it’s not, you might try partially exposing the
roots to encourage more trunks.
Air layering should work great. Sometimes cuttings work good. The last time they all got moldy. If you prune anything off, you might as well try rooting it. I’ve given a lot of plants away.
I’m in Missouri, so my experiences might not apply to warmer climates. Your variety might have different growth habits too.
I have one Hardy Chicago fig outside, planted in the ground about 5 years ago. Until this year, it died back to the ground every year. This year, two trunks only died back to 18 “. Even when it dies back, it grows 8’ tall trunks and produces some fruit the same year. It’s really amazing. I’m hoping the older
wood that survived will let more fruit mature before frost.
I also have one in a 15 gallon
feed tub that I over wintered in the garage. That worked great.