Can I construct a trigger that deletes existing rows before inserting new ones in SQLite, which lacks FOR EACH STATEMENT?
Remember that this is a lite version of SQL and does not support all operations. See the following link:

https://www.sqlite.org/lang_createtrigger.html

SQLite supports a BEFORE TRIGGER which can delete rows before further action is taken. The link also provides some cautions about how that works:


  If a BEFORE UPDATE or BEFORE DELETE trigger modifies or deletes a row
  that was to have been updated or deleted, then the result of the
  subsequent update or delete operation is undefined. Furthermore, if a
  BEFORE trigger modifies or deletes a row, then it is undefined whether
  or not AFTER triggers that would have otherwise run on those rows will
  in fact run. 


See the link to learn other details about the SQLite triggers.

Of course you could write your SQLite code to delete the rows that you no longer need in your database before inserting new rows into your database. I would prefer that approach over a trigger since your are just getting rid of unwanted data.