If you have trouble keeping the bowl still, there are several techniques you can try. A slightly damp towel under the bowl may suffice. If that doesn’t work, try a piece of grip texture shelf liner or a silicone mat under the bowl. Batter bowls have either a rubbery bottom or non-slip feet to help hold the bowl in position. If you need to use a whisk to beat eggs, cream, or a thin batter, you may need the help of a Swedish preparation board and its vise to hold the bowl firm. If that is not an option for you, perhaps you can use a blender or electric mixer to get the job done.
A hand-held mixer can be used if your mixing bowl will hold still, either by weight or by a captive device, such as the vise on a Swedish cutting board. However, I prefer a heavy duty stand mixer that holds either the bowl or the beaters rigidly. If you plan to make bread dough or mix other heavy batters, you will need a mixer equipped with a dough hook. I use a Kitchen Aid Artisan mixer with 300 watt motor. It seems able for any mixing task. Kitchen Aid sells several models.
I can manage my Oster blender with one hand. The only parts that are difficult to separate are the glass chamber and the blade base. My normal way of dealing with this is to (1) rinse the assembled piece immediately after use then (2) add a few drops of dish washing liquid and 1-2 cups of water to the blending chamber, (3) mount the blending chamber and blend the soap-water at high setting to thoroughly wash the chamber, and (4) thoroughly rinse the clean chamber. If that process is not adequate, I either call for help or use my Swedish board’s vise to remove the blade base. The model of Oster blender that I use doesn’t seem to be on the market any longer, but there seems to be many that can be managed with one hand.