Well this is weird.
Your machine probably has a neutral drain. What this means is that the transmission is in neutral when the drain function works. After several minutes the timer will stop the motor, which causes the transmission to shift into spin mode.
If the timer is not doing this stop the transmission never shifts. What you are doing when you open the lid is pausing the motor, causing the transmission to shift and then it will drain.
Replacing the timer should have fixed this, that's the weird part. I would check the connector pins and seating on the timer, clean everything with a good contact cleaner and then see if that fixes things. Probably will not but worth a try.
I would then try replacing the timer again. If you buy from this site and the part does not fix the problem you can return it.
Other than that, no good ideas.