Can a Mujtahid Muqayyad pass rulings on the basis of the methodology of different schools of thought on different legal issues?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

The mujtahid in a certain Mathhab (school of Fiqh), also known as a Mujtahid Muqayyad (i.e. a restricted Mujtahid), is the scholar who deduces rulings based on evidence according to the methodology of the Imaam of his Mathhab, and he deduces different rulings from the narrations stated by his Imaam. [Commentary of Shaykh ‘Abd Ar-Razzaaq ‘Afeefi on Al-Aamidi's book Al-Ihkaam]

That said, the Mujtahid Muqayyad is not allowed to make deductions on the basis of the methodology of other than his Imaam or to deduce different rulings from the narrations of his Imaam until he is completely knowledgeable about that.

It should be noted, though, that the preponderant opinion among scholars of Usool Al-Fiqh (principles of jurisprudence) is that Ijtihaad can be divided (i.e. the independent reasoning of a certain jurist can be divided into the two different categories of Ijtihaad). Hence, a scholar may attain the rank of absolute Ijtihaad (i.e. to be a Mujtahid Mutlaq who fulfills all conditions of Ijtihaad) in a certain issue or branch of Fiqh and accordingly becomes legislatively entitled to issue his independent fatwas on it, even if he is a Mujtahid Muqayyad in other than it.  

Ibn As-Salah wrote, "The Mujtahid Muqayyad may be qualified to practice absolute Ijtihaad in a certain issue or a certain branch, like the Mujtahids of the previous category."

Allah knows best.