Very carefully? I used to work on trawl surveys in Rhode Island where we would occasionally catch large Atlantic Torpedo Rays (T. nobiliana). We used a hard rubber or plastic shovel to lift and release them.
The thick rubber mats used in the foot wells of cars are really useful for this purpose! They are flexible enough that you can trap a ray beneath one to keep it contained, and you can cut a hole in the mat to allow a safe tagging spot: simply manoeuvre the hole over the area you wish to insert the tag. The mats are available in every size imaginable so if you got a large size to place the ray on, and then a smaller one to hold on top of it, the whole lot could be lifted up and tipped over the side to release the ray at the end of tagging. Hope this helps!
Some towels on the rubber mats do help to avoiding 'puddling' of water which can be an issue with these species. Getting them out of the nets might be tricky. If you have enough help, try to make the net taught and let the ray land on the rubber mat. This is going to be very difficult if the rays get tangled in the nets to any extent, and you will need to have some extra thick industrial strength rubber gloves on hand to be able to help remove any tangled rays. I wish you luck!!!