If you have an existing php application, then there is no point to migrate it to some framework. Frameworks help you to develop project faster - you already have one. At this point adding framework on top of what you have will only make your application more complex and, most likely, slower.
Instead of trying to find some magical framework, you should invest in refactoring (apply some OOP principles, like SOLID) and documenting the existing code. Maybe start incrementally add some unit-tests.
P.S. Doctrine is an ORM ( 1.x implements activerectord , 2.x - something like a datamapper ). ORMs are for people who are extremely lazy, misguided or do not understand SQL. By moving your existing SQL queries to ORMs code you will make it all only slower. Additionally, you will encounter problems when migrating complicated queries. Just dont do it.