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.

Form and function are inextricably linked: one will inform the other. A lot of the ergo-split community focuses on the use case where you move your hands as little as possible, and the designs tend to revolve around maximizing that ideal. And they are damn good at it. The drawback, as you note, is that it's a design that expects you not to move your hands around: it encourages keyboard navigation and shortcuts in place of using the mouse as much as possible.

That said, you can get around it. You can use layers to move common shortcuts to the left hand, so you don't have to do the whole "Stretch my hand across two units" dance. Or, you can look into something like a macro pad.

Me, I just deal. The comfort when typing is well worth the tradeoff, to me. I'll favor avoiding the mouse when possible, and just dance my one hand across both halves when needed. It's not a huge deal to me, but the whole point is personalization: find what works best for you!

You could try out QMKs swap hands feature. I've not tried using myself but have considered doing so for this use case exactly.

I moved my mouse between the two splits.

I am a vim user (and use vim keybinding for web browsing), so I rarely need to use both my mouse and keyboard at the same time.

If I need to type with one hand I just push them back together. Or just use the second one piece keyboard that I have for gaming use.

Single handed operation is usefull for my use case as well (mainly CAD and graphic work). I've tried various approaches:

• Swap hands feature from QMK: it works, but has some limitations. I also just made a layer with swapped layout to work around some of the limitations, but ultimately I just didn't find it very convenient.
• I then proceeded to write my own "autoswap" feature, inspired by autoshift. Basically it is swap hands by holding any alpha key. So tap Q is Q, hold Q results in P. That worked brilliantly, but the compromise is that you can't use any tap-hold functions on your alphas (like layer taps or homerow mods). Since I don't like homerow mods anyway, I could manage with my 36 key layout, but it was definitely a compromise to fit all mods and layer taps on thumbs or combos. Going up to 40 keys would probably have made this a pretty decent solution though.
• Then I added a trackball to my keyboard. Having the trackball integrated still required me to move my right hand slightly away from the home row, but its much less of an issue than before, and it made any swap hands feature redundant. I definitely recommend looking into something like a Charybdis, probably the best straight out of box solution (out of box solution being relative ofc in this mad world of DIY ergo boards).
• Right now I'm experimenting with a layout inspired by TAIPO; its a chording layout, that is intended for use with 2 hands on a 20 key layout, but left and right are mirrored, so it is fully functional in one handed operation as well. Its definitely a learning curve, but I'm surprised by how natural it feels. Its like a 10 key macropad, which is somehow magically equally functional as a full keyboard, without feeling like a compromise! I definitely recommend at least reading up on it: https://inkeys.wiki/en/keymaps/taipo ; it may be a bit much to get into if you are new to ergo keyboards and small layouts though, so I would suggest first getting familiar with something like a 36-40 key split and toying around in QMK.

Ok so I use a split since two months and the switch is certainly not automatic but the solutions are fairly simple. I’m a good exemple of a user scenario that requires a lot of one-handed operations because I mainly edit text and not writing it and I use a lot of GUI related to graphic work (illustrator mainly but also photoshop). For the following process, I should make clear that I’m on a Mac and it’s clearly a big plus since this platform promotes consistent patterns of shortcuts.

Text edition :

• I have a ⇧ and a ⌘ key on my thumbs and arrows on a layer triggered by a thumb. Lateral arrows are set to trigger ⎇+arrow when long press and ⌘+arrow when double presses. With the ⇧ on the right thumb and the arrows (and the layer switch key) on my left hand. Give it a try, I can really quickly select words, lines, etc. For a quick word selection/replacement I have also a simple macro that trigger : “⎇+⬅️, ⇧+⎇+➡️, ⌘-c” (the copy part is just in case I want this bit of text in my pasteboard.
• I use vim whenever I can. It makes the use of a mouse irrelevant, don’t worry the main selection operation are easy to learn.

GUI

• It’s more tricky but I really advise to take a note of every friction in your workflow and find a solution when mandatory. I could give some of my settings but it’s very specific to the app I use. The only advice that could match any situation is a “esc” signal that is always accessible as a combo with the W+E combination, in every layer, I also have a space and I try to get the maximum access of elementary keys but when it replicates a primary action on my right hand, I hide it on my left hand (combo, double-tap)

I should mention that I use Colemak and that it certainly help compared to layout leaning toward hand switch (Dvorak, Bepo…). But if you’re on QWERTY it should be good.

You could use autohot key to setup a FrogPad like meta key to give you access to all the keys from one side. https://en.wikipedia.org/wiki/FrogPad

I've been using a Corne for over a year. The halves are close enough that I can move my non-mousing hand to the other half. I like the split primarily for a more comfortable typing angle rather than a wider stance.

When I started using home row mods I also experienced issues with shortcuts requiring ctrl (copy, paste, cut, dave, select all, close window, new tab, erc)

The solution I found was to map Ctrl in multiple positions only on the left hand so I do not need any contortions

I have a Moonlander, and use the keyboard itself for most of my “mouse” needs. You can do the same on the Sofle: https://github.com/qmk/qmk_firmware/blob/master/docs/feature_mouse_keys.md

I also have trackpad between the keyboard halves for scrolling while eating or when I need to make ultra-fine selections. For gaming I use the left half of the board and a mouse in my right hand.