Dear Kate,
I understand your situation and I do not think there is a technical solution to it. If you do change it to manual self completetion then the students can 'cheat' unless a teacher operates a separate tick sheet to confirm through questioning that the student has completed the task before they click on the completion and any ticks without teacher confirmation is cheating.
The issue here is running two parallel systems, IMHO either you operate the online system or the classroom system for all.
However, you could choose to ignore activity completion and set an end of course / topic test as the metric of assessment. And any students who did not complete the tasks required of them will get lower marks.
Cheers,
Gareth