If you want to perform the migration on database level, I would recommend to add a new column to your post table so you can first create a copy of your old data and then compare the results of a migration script. After adding the new column, you could use REGEXP_REPLACE as part of an UPDATE to copy the data into the new column. You may have to change the regular expression syntax so it catches all characters you encounter in the "style" section but this could be a start:
UPDATE
post_table
SET
stripped_text = regexp_replace(original_text,
' style="[[:alnum:][:digit:][:space:];,:\(\)\'\-\.]+"', '')
Examine the data and see if the results are OK, repeat until the result is good enough. Rename your original post column ("original_text" in my sample) to something like "orignal_backup" and your new column ("stripped_text" in my example) to the original post column's name.
More information regarding character classes in MySQL REGEX can be found here.