There are a few different options, though none of them are without trade-offs.

1. You could try getting a pointing device integrated into your keyboard, like a trackpad, trackpoint, or trackball that could be used with a single finger while the rest of the fingers are more or less still in the home position. For example, a tractyl.
2. You configure your left hand side (assuming you mouse with the right) to have extra layers which give access to a full keyboard from just that hand. The first thing I would do here is figure out how much you really need to access... Do you just need copy/paste/cut? Do you just need alpha keys? Do you really need a full keyboard including symbols and alt, F-keys, etc? For an extreme version of this, this keyboard using a Taipo layout gives either hand access to a FULL keyboard with only 11 keys per hand, and you have almost 3 times as many keys to work with. You can type entirely with just the left side or entirely with just the right side with that keyboard, though going that small will have a steep learning curve.
3. Get something with even more keys than a sofle, but for just the left split, and you'd only use those extra keys while one handed typing, so each one of those keys is a duplicate function that can be achieved in a different way when you're using both hands. You could even use a 60% keyboard as your left split, as it would mostly just invade the space between the keyboards which you might have free due to the split.