Is it possible to integrate paired-end sequences with different primers?
Yes, but there are some limitations to be aware of.
If you want to have shared ASVs across primers, then yes, you will need to trim to the exact same region across all three runs.
So you can choose to trim to the same region to get one unified ASV table, or you can process all three samples separately, and end up with three ASV tables.
You are not the first person to have this problem! Check out this thread where another teams is working with data from multiple variable regions.
Let me know if that helps answer your question!
Colin