Easy. I use a Stryd footpod and a Wahoo TickR chest strap. I do not use the data from the treadmill.
The treadmill can receive and display heart rate from an ANT+ strap. As far as broadcasting data, I think it’s only Bluetooth, which means one connection at a time, so if Bluetooth is going to Zwift it can’t also go to the watch. But you’ll only get speed/distance from the treadmill, nothing else.
Stryd (with ANT+) avoids all the hassle of Bluetooth limitations and, of course, gives me power data, which the treadmill will not.
p.s. The above setup is how I use Zwift with a laptop and ANT+ dongle. If I use a tablet instead of laptop then I would need Bluetooth outputs from Stryd and the chest strap for Zwift. The ANT+ signals from Stryd and strap can still go to the watch at the same time.
I’ve also noticed that with the tablet option I cannot use power data from Stryd to move a bike on Zwift. On laptop I can use Stryd power to move a bike on Zwift, which for me is a more involving option.