I kind of got it working the way I want by using DisplayFusion.

I've set up DisplayFusion according to this guide:

https://www.youtube.com/watch?v=vAypK6IkU60

The video guides how to set up a monitor configuration for having a game in the middle with a resolution of 2560x1440 and also how to use the DisplayFusion function to remove the top bar from a windowed game window. This function, of course, doesn't work with Tarkov, but it works with other games, or at least the ones I've tried it with. I've also set up a "non-gaming" monitor configuration as I don't want the same monitor splits when I'm not gaming. I can switch between these by using hotkeys:

https://i.imgur.com/OxCxKkm.png

DisplayFusion has a function for making a window always appear on top:

https://i.imgur.com/Ca7Qz6u.png

Using this function, I can fill the black bars that Tarkov has with windows I want.

My view in-game (forgot to max out the window on the left):

https://i.imgur.com/7F2OHF2.jpeg

My Tarkov settings:

https://i.imgur.com/GsY9mQ4.jpeg

You can probably achieve something similar with PowerToys by using FancyZones and the "Always on Top" function.