Not sure if I understand the problem here.
Please let me know if the following suggestion would solve your problem:
Let’s say, for ex. you have your mic on INPUT1, you send LINE-OUT-1L to your ext. effect and send it back in to INPUT5.
You use the UR’s MIX1 to send only the dry INPUT1 signal to LINE-OUT-1L.
You use MIX2 to route the wet (IN5) to LINE-OUT-2L (and PHONES2).
In your DAW, you can record both IN1 (dry) and IN5 (wet) separately.
You might need later to nudge the wet track to align perfectly with the dry signal in your DAW if your external effect gear added noticeable latency, and only if you want to use both in the same mix of course.
Connecting your headphones to PHONES2 jack, you can also quickly toggle wet-dry by selecting MIX1 or MIX2.
Monitoring on either the PHONES2 jack or LINE-OUT-1 or -2 is done without any latency (not going through the DAW).
PHONES2 jack can be used by the artist with the wet signal.
PHONES1 jack is sending the dry signal to the recording eng.
Many many other routing and wet/dry combinations are possible of course.
You can also send a click track and music to MIX2 for the artist to sing to.
If doing so, some track nudging might be needed to realign everything (to perfection…) after recording.
Hope this helped…