I think there are really two options for this. One is to just do them simultaneously and not worry about it. As mentioned above, careful placement of your lines can help mitigate this problem. I have my new water delivered into the return pump chamber in my sump and the old water removed from the chamber in the sump where the tank drains. Since my AWC runs 24/7 it only changes 66ml every 10 minutes, so it’s very unlikely that any of the new water is making it through the DT and being removed with the old water.
If you don’t want your ATO running continuously and you want to change a lot of water at once over a short time period then you are more likely to run into an issue where you are removing some new water with the old water. In that case you can drain first, then add and simply have the apex turn off your ATO and rollermat while the AWC happens, then turn them back on when it’s complete. If you need help with the programming for this, let us know.