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How are all of these things connected? Please detail each connection. Headphones connected via wire, mic built into the PC, the phone is connected to... etc. etc. etc.
How are you switching these things? Via controls on the phone, on the PC, something on the mic or headphones?
You talk about the mic as if the headphones do not have a mic of their own, do your headphones have a mic?
I saw your post a few days ago and wasn't sure if anyone would attempt a tip or two because this isn't really an Android question other than it sounds like you're playing the game on your phone.
Without knowing anything about your set-up it sounds like you need headphones with a mic. Switching to whatever external mic you're using seems to disconnect or take the place of the headphones as an external device. Headphones like Turtle Beach have adjustable independent mic levels from the headphone level and offers the ability to mute the mic locally on the headphones which might allow you to keep the mic hot and connected, enabling it only when you need it. It also sounds like you might be able to achieve this result if both headphones and mic were connected to the PC in the same way, so using an external wired mic and wired headphones that plug into jacks on the unit, or having both wireless connected via BT, because it sounds like you're mixing connection types and switching between them is causing this disconnect of the other. I don't use sound devices connected to a laptop or anything like that so really I'm not an expert on any of this and without answers from you on how these things are connected and which have what these are all just guesses from an Android guy who isn't familiar with any of this.
Honestly this would be more of a question for other gaming streamers who are doing this kind of thing. A quick search turned up these two steamer forums...
www.gaming-asylum.com
I'm more than willing to help you stumble through this but you'll probably get better help from those who know at one of the other forums, but answer the questions posed and we'll go from there.