We're running into issues here on a VDI build-out with the SpeechMike as well, also running Horizon 6.2, and I'm reaching out to Phillips for assistance since our issue is that the SpeechMike isn't functioning even when it is properly passed through to the VM. The first thing I want to note on your experience is that it sounds like RTAV is picking up your speechmike and is handling the audio in/out. You can confirm this by checking what is passing the audio on the VM, if you see "VMware Virtual Microphone" as the audio detection source, then it's VMware's Realtime Audio-Video (RTAV) that's passing that audio and not USB redirection. The trackball will also work by default because if it works on the physical then using it moves the physical machine's pointer and well... there you go, that's obviously being translated through the Horizon session like any local mouse movement (by the same token, the mouse-click button on the bottom of your SpeechMike should also work).
As to why you're not seeing the device for USB redirection, you may be missing the GPO settings for splitting the SpeechMike, which you'll need to do if you want RTAV for the audio, and to still pass the button functionality through USB redirection. You can also simply use the the GPO to "include" instead of split and pass the entire SpeechMike through USB redirection. However, the reason you want RTAV for the audio is because it keeps the audio from utilizing a large amount of bandwidth (SpeechMike is ~1Mbps according to the Nuance engineers I've been working with). If you're deploying these at a large scale 1000+, or to any remote sites or other locations where bandwidth is limited, I'd definitely recommend you utilize RTAV or another solution (such as Nuance's VMware Client Extension) to reduce your bandwidth needs.
Here's a KB regarding the Vid/Pids: Nuance SpeechMagic VMware View Extension - Supported USB Devices and USB Configuration (2068447) | VMware KB - Unfortunately that's specific to SpeechMagic and an older version of Horizon, but the split info is still relevant here, since the functionality of RTAV is very similar to that of the SpeechMagic extension. So, the VID/PID you're looking for with the 3200 is 'vid-0911_pid-0c1c'. Of course you need to have the vdm_agent admin template installed, then I'd recommend making the following changes to the GPO applying to your VMs:
To pass through the entire device using USB redirection:
Enable the following GPO and set its value to 'o:vid-0911_pid-0c1c':
Computer Configuration > Policies > Administrative Templates > Classic Administrative Templates > VMware View Agent Configuration > View USB Configuration > Include Vid/Pid Device
To pass through the button functionality and leave the audio for RTAV:
Enable the following GPO and set its value to 'o:vid-0911_pid-0c1c(exintf:00;exintf:01;exintf:02;exintf:03)'
Computer Configuration > Policies > Administrative Templates > Classic Administrative Templates > VMware View Agent Configuration > View USB Configuration > Client Downloadable only Settings > Split Vid/Pid Device
After configuring those you should be able to redirect the speechmike via USB redirection. This is where my issue begins, I can split the speechmike and redirect it, but it doesn't function in the VM with or without the Phillips SpeechMike drivers installed. The only part that works is VMware's RTAV, just like your situation. If I redirect the entire SpeechMike, I have the same issue, the audio works via the USB redirection now instead of RTAV, but both with and without the SpeechMike drivers the buttons do not function. Hope my info on this is helpful, sorry for the wall of text.
~LikeABrandit