Perhaps this is no longer an issue for you, but…
It is possible to ride on Zwift without copying the workout over if your devices have both ANT+ and Bluetooth capability. I connect to SYSTM using ANT+, then I start Zwift and use the Zwift Companion to connect my devices to Zwift using Bluetooth. I then unpair the controllable trainer pairing on Zwift because I want SYSTM to do that.
I do this a lot. Just did that this morning and on Monday. I ran my no-vid on my phone connected to BT and controlling my power and then ran Zwift on a nearby computer with an ANT+ dongle. Since the Kickr has multiple BT channels I actually had them both connected thru BT, but I used ANT+ to connect my HR and cadence to my PC. Since it was an easy ride I just picked a flat route on Zwift and did loops. And then I listened to music from my phone thru my BT headphones. Then my son woke up and walked in towards the end. Full sensory overload. Lol.
Sometimes I switch and put Zwift on my phone and the no-vid on my pc with the big screen. It depends on what I want to focus on.
Actually, on Tuesday I had Zwift on my phone, the no-vid on my pc in MiniPlayer mode, and watched some races on GCN in the background. Not sure why I needed Zwift running other than to build up my stats. 
