Bypass MyISAM table lock
Since your main issue is the FULLTEXT index, I have two storage engines suggestions
SUGGESTION #1 : MyISAM
The MyISAM storage engine does a full table lock on the following:

ALTER TABLE
DELETE
UPDATE
INSERT

The only exception to the table lock blocking everything would be setting concurrent_insert. (See MySQL Documentation on Concurrent Inserts)
Try setting this in my.cnf
[mysqld]
concurrent_insert = ALWAYS

or
[mysqld]
concurrent_insert = 2

Adding new rows to a MyISAM table bypasses table locking only in this circumstance. You will have to experiment with adding rows to the table while updating fulltext indexed columns in another row of the table.
SUGGESTION # 2 : InnoDB
As of Feb 5, 2013, MySQL 5.6 is not out of beta and is now GA. InnoDB in MySQL 5.6 supports FULLTEXT indexing. Since doing this in an ACID-compliant Database is what you essentially need, look into upgrading. Do no load any preinstalled MySQL 5.6 packages if they predate Feb 5, 2013.