What does the phrase "with suspect allegiances required to combat a powerful enemy" mean exactly here?

That text could do with some commas to make it a little easier to parse, IMO. Eg

"It would rewrite Henry V by offering another version of the story, of an ambivalent heroic warrior with suspect allegiances, required to combat a powerful enemy, who is pushed to his physical and emotional limits, and marries outside his community."

So, the segment you asked about is now part of two different phrases:

It would rewrite Henry V by offering another version of the story,

of an ambivalent heroic warrior with suspect allegiances (the warrior's allegiances are suspect, ie under suspicion)

and that warrior experiences the following things:

1. he is required to combat a powerful enemy
2. he is pushed to his physical and emotional limits
3. he marries outside his community