The ultimate solution was to write a query to pull the data out of the table and insert it into a new table as CLOB (which SSMA correctly deals with automatically as varchar(MAX) and doesn't truncate) via a built-in Oracle function.
I used a command similar to the following in sqlplus:
create table schema.my_new_table as
select primary_key_field, to_lob(troublesome_long_column) troublesome_long_column
from schema.original_table;
Since I included the primary key fields, I can tie each row back to the original table. The "to_lob" function converts LONG to CLOB, which makes the conversion of the data in the new table work.
I would have altered the original table (and just added a new column with the converted data), but I couldn't change anything about the existing tables.