Bluee asked me to post a tutorial on how to get your game window to still fit fully in your screen even though it's outputting at the higher resolution that you told it to with SourceRes. Side effects include mouse detection being misaligned in-game (aka you can hardly use the mouse), horrible downscaling (very pixelated because it uses a nearest neighbor technique; you can see the before and after (native and shrunken w/ SourceRes, respectively) here and here. Those screenshots are simulated but it gets the point across; in practice it'll probably look even worse.
Anyway, you need to install GameCompanion from here. This utility is used for many other games in order to force a borderless window output mode. Essentially it strips the borders from a window and then stretches it with a form of overlay to fit across the screen. This will stretch with nearest neighbor techniques so if you output to a resolution that isn't your native res (usually lower, but in our case, higher), the result will look atrocious, as shown earlier. After installing GameCompanion, open it and find out what your "Manual FFMode" hotkey is. Mine is Ctrl+Enter, but I don't recall if this is the default. Anyway, start TF2 now with the "-windowed -insecure" parameters (additionally "-novid" or any other parameters, but do not use "-noborder", as I'm not sure how it behaves with GameCompanion). Once it's launched in a window, use SourceRes in the console to target the output resolution you want - for example, to make your game output at 4K, you'd enter "sr_forceres 3840 2160". Then, once it's adjusted your resolution, hit Ctrl+Enter (or whatever your hotkey is/was) while still in the console. Staying in the console is entirely optional, but is definitely more convenient as the "enter" key will probably trigger a menu button, and you'll want the console open afterward anyway. This should stretch your window to fill your primary monitor.
From here, you're pretty much done. You're viewing a downscaled version of the incredibly high res game now, but even though your view is downscaled, all images and videos will be rendered at the SourceRes resolution. You can close GameCompanion when you're done capturing. Enjoy!
EDIT: In order to reassure based on previous complaints, mouse tracking should only be broken for navigating menus and things of that nature. Rotating for thirdperson shots should be okay.
EDIT2, 10 months later: For those of you trying to get this to work, be sure that in GameCompanion's settings window, in the Options tab, you have FFMode set to Type2, not Type1. Be sure to restart both GameCompanion and TF2 fully once setting this option for it to take effect.