It is likely that the delete statement will affect a large fraction of the total rows in the table. Eventually this might lead to a table lock being acquired when deleting. Holding on to a lock (in this case row- or page locks) and acquiring more locks is always a deadlock risk. However I can't explain why the insert statement leads to a lock escalation - it might have to do with page splitting/adding, but someone knowing MySQL better will have to fill in there.
For a start it can be worth trying to explicitly acquire a table lock right away for the delete statement. See LOCK TABLES and Table locking issues.