I own or have owned all the ones I linked below. I don't use my laptop as a desktop so it's not connected to any of my monitors. The reason I bought the stands is to house the laptop when not in used or when I connect it to my TV.

1. iQunix Vertical Laptop Stand: This laptop stand works well. The rubber padding is pretty thick so I didn't worry too much about possibly scratching it, especially around the opening edges. Also, the design of this stand allows good airflow if you want to have the fan vent pointing down.
2. Twelve South BookArc: This stand is probably one of the nicest stand I owned. It's everything above plus the leg slot that allows you to run your cable under if needed. If you're looking for a stand to display your laptop, I say this would be it. Also, because of the extra rubber padding around the opening of the stand, you can slide the laptop in with ease.
3. Jarlink Vertical Stand: This here is a decent stand that I liked because it's pretty minimal. It's smaller than the other two and has a nice weight to it. It's also adjustable so it should work with most laptops. However, it's kind of a PIA to adjust the thing to the right fitting if you have OCD like me. Oh, there are no slots on the bottom so ventilation isn't great depending on how you use it.
4. Seenda Vertical Stand: This is the stand that I use the most when I'm not using my desktop. It's adjustable to fit most laptop sizes and can be used as a height stand for a more comfortable typing position (I use the height stand a lot). It also has a 2nd slot where I place my iPad (this was a huge plus for me). There are no slots on the bottom so ventilation isn't great depending on how you use it. The biggest con to this stand is that there are no rubber padding around the top edges of the opening. So when you slide your laptop in, you want to be a little careful.

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I was in the exact same situation - dual monitors for my PC, but I also use both monitors for my work laptop during the day. Here's what I use:

$25 KVM from Microcenter- I only use this for my keyboard since I have separate mice for work/home, but you could use this for both keyboard and mouse

$17 Audio switcher from Amazon - I have 3.5mm headphone jacks coming from both my laptop and PC into this, then switch based on which machine I'm working from.

$20 Vertical Laptop Stand - this is to keep my laptop behind my monitor so it's tucked up out of the way. You don't need to spend a cent more than you need to on these, unless the laptop goes in and out a lot, just get a cheap one.

My work laptop stays behind my monitors, but I have a Dell USB-C hub that outputs video via 2 cords - 1 HDMI and 1 VGA

Both of my monitors (and most out there) should have more than 1 input, hopefully it has something like Display Port/HDMI/VGA. I have my PC sending video via DP to my left monitor, and HDMI to my right monitor. My laptop dock outputs HDMI to my left monitor, and VGA to my right monitor. This last part is super dependent on what monitors and what USB hub you use. The only issue here is if you use HDMI outputs for both monitors from your PC and work laptop, you might run out of input ports on the monitors.

Let me know your setup and I can try and tailor suggestions, but this setup is what I've come up with after tinkering with stuff for the last few months.

I have option 2 and I love it but I would also add something like this. What are the models of each of your monitors because I feel like there are more secure options for the vesa mount kit. Also, how are you hooking up your monitors and keyboard and mouse to the computer?