What an interesting idea that Edward is a tragic hero! In reality, I think either Brian OR Edward could be considered a tragic hero. Why not look at both? (I actually think Brian is a better fit for the tragic hero type, but you can be the judge after we look at the evidence.) A tragic hero is generally honorable, but has a flaw and ultimately meets his fate; therefore, this could describe either Brian OR Edward.
In regards to Brian possibly being the tragic hero, he denies his true homosexual nature by conforming to society's norms (and his aunt's prodding) by pursuing Philippa Archibald (at least for a while). It is this flaw that causes Brian's true love, Edward, to join the army and go away to fight in the Spanish Civil War. At this point, Brian realizes his mistake, admits to Philippa that he can never be a heterosexual husband, decides to "rescue" Edward from war, and travels out to sea to find him. It is here that Brian meets reality: his lover has died due to typhoid fever that he has contracted at sea. If Brian had admitted his true nature at the beginning, embraced his homosexuality, thwarted his Aunt Constance, denied Philippa, and married Edward, Edward and Brian could have spent a long lifetime together instead of being met with an early demise.
It is Edward being the tragic hero that I think is the most interesting! Edward's flaw is not the same as Brian's. Edward is happy to embrace his homosexuality. Edward's main flaw is fleeing the situation without much thought to the consequences. Fleeing INTO WAR is not exactly a safe prospect! Sure enough, Edward dies as a result, although not in the way that one might think. Edward doesn't die in valor on the battlefield. No. Edward eats contaminated food and lives amid filthy conditions; therefore, he contracts typhoid fever and dies.
Note the following quotation:
And at that moment a happiness filled me that was pure and perfect and yet it was bled with despair - as if I had been handed a cup of ambrosial nectar to drink from and knew that once I finished drinking, the cup would be withdrawn forever, and nothing to come would ever taste as good.
Both characters (Brian by denying his homosexuality at first and Edward by fleeing into war) decide, in some way, to reject this ultimate love. This could be said to be the general tragic flaw of BOTH characters.