So the problem is that a large table's backup is too large to download, even if compressed? That would mean that you would have to split the table into several pieces. One way would be to fix and modify the SMF backup to split large table into multiple pieces, saved under different names. You could use MySQL queries to do SELECT * FROM {prefix}{tablename} LIMIT 1000,{starting record 0, 1000, 2000, etc.} ORDER BY {some field such as an ID} and write into file backup_{tablename}_{starting record}.sql.
I don't see anything in phpMyAdmin > Export that will let you select a section of a table, nor does there seem to be a clean way to copy just part of a table to another table. It would probably cleanest to write a PHP script to dump a table in sections. Of course, if you start with the SMF backup code, that means taking care of all the little problems in it so you get a clean backup.