Beren was a Hero
Firstly, in the Legendarium, everything happens with accordance to the Music, fate is the main driver in the stories of Tolkien's Legendarium. More info on that can be found here but for now I will just say, if fate had not been the main driver in Tolkien's Legendarium the Narn i Chîn Húrin (Tales of the Children of Hurin) would've taken a very different turn.
Back to Beren:
A hero in Tolkien's world is often one who adheres to the good, the will of Eru and Manwë, further, one who achieves greatness through purity. Beren can be considered one of the Greatest Heroes, having been able to break through the Girdle of Melian, a feat that not many could do, save Morgoth and Turin (who can be considered a doom-filled hero).
Beren however was on a mission. Thingol sent Beren to reclaim a Silmaril if we wanted to take his daughter's hand in marriage.
“For little price,’ he said, ‘do Elven-kings sell their daughters: for gems, and things made by craft. But if this be your will, Thingol, I will perform it. And when we meet again my hand shall hold a Silmaril from the Iron Crown; for you have not looked the last upon Beren son of Barahir.”
The Silmarillion - Chapter 19, Of Beren and Luthien
However, after Morgoth falls from his throne in slumber, Beren, tasked with taking the Silmaril, has no other thought but to take the Silmaril and fulfil his task. However after Beren gets greedy and strays slightly from his fate he is doomed.
As he closed it in his hand, the radiance welled through his living flesh, and his hand became as a shining lamp; but the jewel suffered his touch and hurt him not. It came then into Beren’s mind that he would go beyond his vow, and bear out of Angband all three of the Jewels of Fëanor; but such was not the doom of the Silmarils. The knife Angrist snapped, and a shard of the blade flying smote the cheek of Morgoth. He groaned and stirred, and all the host of Angband moved in sleep.
Then terror fell upon Beren and Lúthien, and they fled, heedless and without disguise, desiring only to see the light once more.
The first emphasised part shows that Beren became selfish, and from the light of the Silmaril was swooned into attempting to steal the rest. However, as always in Tolkien's Legendarium it was not the fate of the Silmarils to be taken by Beren, and therefore Angrist snapped and Melkor was stirred from his sleep.