The small Singer sewing machine you have will struggle
I don't think he said his singer was small, but he did say it's old.
I have three sewing machines. I do a lot of tailoring of suits and shirts, etc. A hobby I picked up a few years ago. I have my mother's 1951 machine, Model 15-91, which is a beast, very strong, but doesn't zig-zag. I have a gorgeous Singer 503a Slant-O-Matic, "the Rocketeer," which is, I think, the pinnacle of Singer home machines, 1960's vintage. It's all mechanical, all steel gears save one big one on the drive wheel, and goes through Sunbrella like nobody's business. And, I have an Alpha Sew walking foot machine, that is the same Chinese machine that is the basis of the Sail Rite machines; mine's a non zig-zag model, though.
I think the 503a can handle just about any sailmaking or canvas making project. I've made some test stitches on Sunbrella using the correct thread (Superior Threads® - Solar Guard #92 Bonded Polyester), and it works very well. It can do 22 different stitches, plus I have the buttonhole attachment.
The old singers are great. And there are industrial ones easily available, too.
On a Bimini you will sew through up to 8 layers.
Thinking about my bimini and dodger I can't imagine anywhere where there are eight layers of fabric stitched. Maybe I should look more closely? Can you give an example of where this might occur?
I think any of my machines would struggle with eight layers of Sunbrella.