i would look for a suitable KVM switch and put that between the dock and your interface devices.
not sure if there is a KVM that works with multiple monitors but if not perhaps you could run a single monitor through the KVM switch and share that with the work computer.

Oddly enough, I HAVE an old KVM switch! LOL I bought it in like 2002, though, and it's been sitting in my closet unused.

It's an OmniView 4-Port USB KVM Switch, model # F1D104-USB :

[kvm-switches-online.com...]

Having never used one, I don't quite get how it works. Surely I don't just plug a USB in to the tablet and then plug the other end to the switch?

No, no, wait... it looks like that KVM switch is going to have a single VGA monitor output, then plugs for a keyboard and mouse or 4 USB outputs.

That's not going to quite work. The tablet we have now has one HDMI output to a monitor, then a USB-C (the mini port) to the dock. The dock then has a USB output for the keyboard and mouse, then its own monitor output.

The dock also has a place for a wired network, which I use because it's considerably faster than using the wireless network through the tablet.

So I think that I would need a switch / dock combo... ? One with 2 or more inputs, then 2 HDMI video outputs and at least 4 USB output ports.

pretend like the dock itself is a computer and then connect it to the KVM switch along with the other computer and the interface devices.

Share an internet connection? No, you'll need a switch or an old router. I've used KVMs between 2 desktops sharing a single keyboard, mouse and screen, but never a laptop and a desktop. And you want to share 3 monitors? The laptop will probably only use a single external monitor anyway.

A switch and a KVM, but limited to a single screen for each. I've had troubles in the past with USB KVMs as well.

Just asking ... have your tried the Remote computing app for this purpose?

Well, shoot, @tangor... I just saw 2 or 3 threads where someone had replied but I didn't get the email :-( I've been working on the Greylist for the new VPS, though, so I guess it got lost in the mix.

I ended up buying a GREATHTEK KVM switch, though:

[amazon.com...]

It works perfectly, with the one exception being that ONE computer "forgets" where all of the icons and everything were placed when she switches to the other computer. The second computer remembers, though, so it HAS to be software... except that they're identical brands and models with the same OS and driver updates, and I haven't changed any software that would (err, should) have an impact on that. So I have to figure that out, but otherwise the switch is great.

I have each computer on the HDMI inputs of the switch, then it outputs to the dock via USB. I had to install Dell's DisplayLink drivers so they would recognize the dock (it was already on the original one, so I just installed it on the new one), but everything else was plug-n-play.

I love semi-happy endings! :)

Life would not be interesting without a few glitches here and there. (sigh!)

I have 3 laptops, 2 monitors and a docking station for one of the laptops. Both monitors are connected to docking station.

To work on these 3 laptops at the same time with one keyboard and one mouse I simply connect to them using Microsoft Remote Desktop.

Easy Peasy and no KVM required.