come to think of it I don't think that even that was enough to take Constantinople.
an avar navy... What navy did the avars have pray tell. As for a persian navy in the Mediterranean.
The Byzantine navy may not have been so skilled but compared to the Persian navy or an Avar navy... Eh it was first class. Besides the Sassanids are a land based power not sea based. The avars morso. Pray tell where nomads were able to find competent sailors to crew ships and actually made enough ships to challenge the byzantine fleet which was at this time pretty big. Not likely at all.
Also unless either side has a way of getting past the Bosporus chain they cant touch Constantinople. As for the walls its asb to have an earthquake happen coincidentally at that time when in otl no earthquake or natural disaster hit the city.
Perhaps you could have a Sassanian general defect to the rhomans. He then becomes rhomanized and converts to Orthodox Christianity and earns reputation for outstanding victories against Persian forces. Then he wins Heraclius favor and soon rises to a high position in the rhoman court. Then the arabs invade like otl. Heraclius loses the levant. Said Persian builds a base of support among disgruntled military and civil aristocrats and launches a cou that overthrows Heraclius and said Persian becomes Emperor after marrying a prominent Byzantine Noble.
Lets also say, said Persian general belonged to a prominent and powerful Persian Noble family.
Ergo you have a Persian noble who takes the rhoman throne. That is the best scenario for you if you want a "Persian" Byzantium. Since technically he and his successors are of Persian descent. Plus once Sassanid Persia falls he being a relative of the Persian royal house who survives would inherit the title of deposed Shah. Therefore you have Emperor of the Rhomans and Shah of the Sassanids.