I'm not a teacher, I can't comment on how appropriate it is to request a Statutory Assessment but there are currently two methods (in my Authority, I can't speak for others) of requesting a Statutory Assessment, either the school applies or the parent can speak directly the Authority & request a Statutory Assessment.
Bearing in mind we're now in a new financial year, it may be worthwhile approaching the SENCO again & going through the report with her. There is nothing to stop you from simultaneously contacting your Local Authority & requesting a Statutory Assessment of your child's special educational needs.
If you decide to request a Statutory Assessment, the Authority will write to you & may include a form for you to fill in, they will also write to the Educational Psychology Services (EPS) and the school - the EPS & the sch will be given 3 weeks to submit their information and six weeks from the date the request is made the Authority will have to make a decision about whether or not to proceed. If the school applies, the Authority will have to make a decision six weeks from the date the application is received. You may have gone through this with the parent partnership already, they may also have informed you that a diagnosis is not enough to secure a Statutory Assessment or a Statement.
You must make the best decision you can about your child & her special needs, if you feel that the school isn't taking appropriate action and you feel requesting a Statutory Assessment is the best thing in the circumstances then that is what you must do. Even if the Authority says no, it might just shake up the school & get them to do something, the Authority may also make recommendations about steps they feel the school should take to meet your child's needs.