The problem is that there can be many solutions, all of which could be proved 'correct' (ie they will not fail structurally) in civil litigation.
But that is not 'engineering' in its true sense, which involves tailoring the solution to the resources available, be they materials, labour, skills, time, access etc.
As your loading seems relatively light, and the span 5+ m, deflection is probably the critical factor in your case. As suggested by tomfe, a 6"x6"x23 could well do, or an 8"x4". Either of those would be a lot less hassle than the flitch beam.