I have multiple (composite) primary keys on a table and one of them will be auto increment. However, interestingly SQLite allows usage of AUTOINCREMENT keyword
No, I don't think this is possible.
You can create a UNIQUE INDEX which has essentially the same effect as a PRIMARY KEY:
CREATE UNIQUE INDEX pk_index ON "table1"("field1","field2");
Besides, I fail to see the logic of your schema, that is -> if a column is autoincrement and you don't intend to mess with the values manually, it's going to be unique anyway, so it makes a good simple short primary key. Why the composite? You may have good reasons to make another index on the combination of columns, though.