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.

Maybe use an InnoDB table for some of the columns, and use a trigger to replicate the text data to a separate MyISAM table for fulltext searches?

Fulltext is relatively new in mysql, and its implementation is not as feature rich as in some other RDBMs. It sounds like you are running up against some fundamental limitations of mysql, and in the long term, if you expect your application to scale further, it may be worth looking into another tool than mysql. Postgres, for example, offers more control over locking and provides a better fulltext implementation.