Yeah, as Keith mentions OBS is probably the way to go, that’s what I use as well. Lots of how-to videos out there as well, which will help cover off all the steps for you I’m sure.
As another possible solution to try, given that you don’t need vocals, you could use Zoom if you’re already familiar with using that? It won’t be as good as OBS in many respects but if you’re just wanting recording to capture your progress, then it could well be sufficient. Here’s what to do:
Ensure you have the latest version of Zoom installed and are signed into an account (free account is fine).
In the audio settings, select your Katana as the microphone, untick auto adjust mic volume and move slider to 100% then also enable the Original Sound option further down plus tick on High Fidelity Music Mode.
Check the video settings has your webcam selected correctly and tick on HD.
Start a New Meeting by clicking the icon on the Zoom app’s home tab.
Once in the meeting, turn ON original sound using the button in the top left (VERY IMPORTANT STEP!)
Make sure your mic is enabled at you can see a green level indication going up and down when you play your guitar.
Hit the Share Screen button then in the pop-up menu, select Advanced and then double click the Computer Audio tile (to select that option).
Hit the record button.
Press play on your backing track, whether that is in YouTube or an audio player on your computer. As long as that audio is being played by your computer and heard out your speakers/headphones at a reasonable level, it should be captured fine in the recording.
Play along with your track then end the Zoom meeting once done. The recording will then process and, when it’s ready, you can review the track to see how well it captured you.
Couple of points to note…
If your computer is a bit older or not very powerful, your guitar might be too far out of sync with the backing track for this approach to work and you will need to use a different method, like OBS, which will allow you to put a delay on the computer’s audio to match them up.
Assuming the above is okay, then you might have to play around with the level of your backing track to have it match nicely with your guitar’s level. So doing a few test recordings first would be wise before playing a full track (just in case you nail it but then realise the levels were all wrong! )