Não há realmente nenhuma boa solução, exceto concatenar ANDs:
SELECT * FROM myTable
WHERE title REGEXP "dog"
AND title REGEXP "cat"
AND title REGEXP "mouse"
Caso contrário, a expressão regular ficaria assim:
SELECT * FROM myTable
WHERE title REGEXP "(dog.*cat.*mouse)|(dog.*mouse.*cat)|(mouse.*dog.*cat)|(mouse.*cat.*dog)|(cat.*dog.*mouse)|(cat.*mouse.*dog)"
