tl:dr; Based on your observation that it works with Moto but not with LG, my guess is that it is due to conflicting TRRS standards -there are two standards. OMTP which is the older standard and CTIA/ AHJ which is the standard all newer Android devices are expected to follow

Details and solutions to overcome are listed

From TS/TRS/TRRS/TRRRS: Combating the misconnection epidemic by Allan Tépper - ProVideo Coalition which talks of this problem on Android phones

There are two TRRS standards - OMTP and CTI / AHJ

• OMTP is the older standard

  
  
  

  
  

  The older OMTP wiring standard calls for Tip to be used for left audio, Ring 1 to be used for right audio, ring 2 to be used for microphone, and the sleeve to be used for ground. According to my research, this older OMTP standard is found in old Nokia (and also Lumia starting from the 2nd generation), old Samsung (2012 Chromebooks), old Sony Ericsson (2010 and 2011 Xperias), Sony (PlayStation Vita) and the OnePlus One I used to own before migrating to a Nexus 6 for use with Project Fi.

  
  

  
  
  
  
  
  • CTIA/AHJ is the newer standard, and is the standard to be followed for all Android devices, except where legal requirements mandate OMTP ( see 3.5 mm Headset: Accessory Specification  |  Android Open Source Project)
  
  
  
  
  

  
  

  The newer CTIA/AHJ wiring standard reverses the last two mentioned, so it calls for Tip to be used for left audio, Ring 1 to be used for right audio, ring 2 to be used for ground, and the sleeve to be used for microphone. According to my research, this newer CTIA/AHJ wiring standard is used in products from Apple, HTC, latest Nokia, latest Samsung, Jolla, Sony (Dualshock 4), Microsoft (including Surface, Lumia, and XboxOne controller with chat adapter) and most Android phones.

  
  

From this, it appears that your devices are using different standards, though it's difficult to say which device is using which standard , but Moto being the older phone is probably using the older standard while LG is using newer standard.

Pictorial representation (see for discussion on more phones including Apple )

Problems in connection are around
Ground - since mic and ground are different in both standards. See also from SE.Electronics Hacking a TRRS to TRS adapter

Solutions

• HAND TO KEY: Converting an OMTP/CTIA headphone to a standard TRS audio connection explains the steps in DIY project




• Use TRRS to 3.5mm Stereo & Microphone Adapter. Alternatively,




• OMTP/CTIA adapter can bridge the gap between a stereo headset and the wrong brand of smartphone ( bi-directional TRRS Standards Adapter for going between CTIA and OMTP or vice/versa)

Samples of the last two options are linked at the end of blog here

• Using a TRRS to RCA adaptor as explained Add an audio input to Android
  . A similar approach worked , though in a different case How do I use an external microphone with my Galaxy Nexus?

Before you invest money, you can possibly verify by swapping the ground and mic , see this example from SE.Electronics

Edit: Post your edit today , saw This you tube video suggests that Wireless TRRS microphone adaptor worked to fix the wireless problem on LG V 10. I don't have any details on that or why it works but maybe worth giving it a shot considering, it's not expensive