I get a smaller volume loaf from mixing sourdough in mixer - what am I doing wrong?
That's quite normal. Hand kneading aligns the gluten well, and stretch-and-fold aligns it better than other methods, due to simple mechanics. The aligned gluten can hold the expansion gasses better. When you knead by mixer or autolyze, the resulting gluten tends to be directionless, more like a sponge than like shingles. The difference may not be too significant in a bread made with commercial yeast, but if you are using sourdough as your leavener, you can very well end up in a situation where you have less gas production (because you are not using an optimized yeast strain) and less elastic gluten formation (because the overdeveloped texture of the starter shortens the new dough), and then you can experience what you described.
Basically, there is no easy solution to what you described. You can try starting a new starter from scratch, using a different method, and hope that it is rise-happier - but this is a ton of work. You could try using a different recipe that is easier to leaven, e.g. with less sourdough, or with a combination of sourdough and yeast, or using a very leavening-friendly flour, such as pure white bread flour (in case you are now using a dark flour or a combination of grains) - but this will change your bread's taste. Or you can try finding out if other mixers with other hook mechanisms produce a better result with your recipe, but there aren't such big differences between home models. All these things require a lot of effort, and depend on chance to find an option that is better than what you currently have (if such an option even exists). So for practical purposes, I would say you have to decide what is the lesser evil for you - low rise or spending time for kneading by hand - and go with it.