You haven't said how are linking the module to the PC. Let's assume you are using the TD27 via USB and using the TD27 as a sound card.
In which case you have an audio loop back
phone --> bluetooth --> module --> PC --> software --> module --> cans
You need to switch off that monitoring loop...
From your requirement specs, viz, "...electronic drum pad to the drum set for additional percussion voices.... dynamically pressure sensitive " then the DTX Multi12 is the best all rounder to suit
....speaking as someone who has a HandSonic (next to me as I type this), Multi12, Octapad...
Use felt beater (if you like the feel, obs) + the Evans. It's only a bit of cloth - I'm surprised you say it's loud.
I usually use QTs (also just a bit of nylon cloth), Evans EQs or Falam Slams (Kevlar) - never had a noise issue. (oh and I've always used felt beaters over many decades - great...
Nope - the Gen16s are not electronic (triggered) cymbals.
They are just acoustic cymbals with loads of holes in them to make them quieter, then mic'd up to make them louder, then shoved through an FX unit in a vain attempt to make them sound good (and failing ;) ).
The Mutli12 has as load of electronic drum sounds in the box. And 100s of world sounds as you requested.
You can always add more drum sounds.
(there's a also some extra free electronic kits you can download https://europe.yamaha.com/en/support/updates/46154_en.html ).
Has a hat control input...
Here's the "best" and most flexible way to record edrums
1. Module --> MIDI data --> Software (DAW/Sequencer/whatever the cool kids are calling it these days. I mostly use Cakewalk/Sonar - www.cakewalk.com - which is now free.)
2. Split MIDI data to individual tracks in DAW if you want...
Can connect anything with MIDI to a "game". Seek out DTXmania for a game example (not specific to the DTX)
If you want to actually learn, look at https://www.roland.com/uk/products/dt-1/ (again, can be made to work with most things)
And the eternal "what kit"? Question.... well here's my...
You have Yamaha hat controller already ? (which one - bit confusing above, you say "sticks with a Yamaha pedal" ;) ?)
- you can get a Yamaha module for not a lot on eBay if you're being "cost conscious" :) (dunno your budget or location)
Anything from anyone. All just £0.50 piezos in a shell. (pretty much anything will "handle" a double kick)
Or - as it's an ekit - it just as simple and can be more flexible and cheaper to get another pad and another single pedal.
"Alesis or Roland"
Those are two of the possible options. There are many, many more. Yamaha, ATV, 2Box, Pearl etc.
You haven't given use much to go on (bit like asking "Toyota or Ford: I need one for driving")
....so here's the "what kit" post:
What have you...
Get maple sticks.
If you're using hickory or even oak they don't flex as well, so transmit vibes.
Flex: maple > hickory > oak
And use as thin as is comfy for your finger length / hand size.
You don't need specialist expensive sticks. Cheap maples are fine. (I get brick of pairs for £20...