Posting this here as a guide as it's something I just figured out how to do and I found no hard and fast answer on how it's done, I kept running into threads on other forums years out of date talking about registry edits then finding it didn't work anymore and messing around with mic mute/unmute programs but nothing was working for me.
If you're like me and you use a mouse button for push to talk on TeamSpeak (I use Button 4) you've probably run into problems when trying to bind the same key in both OBS and ShadowPlay and numerous other programs.
Step 1 (may not apply).
With my mouse (razer mamba) button 4 is the default back button on browsers so I needed to disable it, to do this I use a program called X-Mouse Button Control. After installing simply set whatever buttons on your mouse you don't want to function to "disable" (see spoiler). That doesn't disable the button but it'll stop doing what you don't want it to do. It is possible to do simulated keys with this program but I've found it a little buggy.
Down a program called AutoHotKey, it's a little program where you can set macros to any button on your computer. After it's installed, right click on your desktop, go to "New" and then "AutoHotkey Script" and name it whatever you like. My script is below, quite simply XButton1 is Mouse Button 4 and F13 is the F13 key. I don't have an F13 key and you probably don't either but it still exists and all the way up to F24. Here is a list of all the keys you can use.
Save your script, right click and run it to see if it works, then right click again and click compile if it does. It should create a new icon on your desktop that looks slightly different, just double click to run it. I recommend to set it to run on startup if it works well for you.
Now go to your program of choice and press that mouse button to have it bound.
Hope this guide helps you in some way!