Did Bellatrix stop doubting Snape's loyalty to Voldemort after he killed Dumbledore?
Yes, I think it helped not just Bellatrix but any Death Eater that was on the fence about Snape’s loyalty to Voldemort. Being so close to Dumbledore for all those years, probably made more than just Bellatrix wonder about the depth of Snape’s loyalty to Voldemort. Dumbledore could be quite persuasive, and he was clever and incredibly talented as a wizard, so yes, by killing the Headmaster, Snape quelled any doubts.
Bellatrix thinks she’s the most devoted servant of Voldemort’s, though Barty Crouch Jr. is right up there, so Bellatrix would see anyone else as being less devoted and less loyal to the Dark Lord, because of her frame of reference. She’s not sane. Narcissa Malfoy, Bellatrix’s sister comes off much more nuanced and moderate, but Bellatrix is the sister who spent years in Azkaban, and that must have messed with her mind. Bellatrix is far from sane, but she remained loyal to Voldemort all those long years in Azkaban.
Yes, when she was sentenced to Azkaban, Bella shrieked about being devoted and waiting for Voldemort’s return, which shows her great loyalty to him, but that was before her prison time. Seeing Bellatrix running free in the last few books, shows us how truly diabolical she is. Bella is scary and only outdone by Voldemort. He’s bent on world dominion and power, but also eternal life, and Bellatrix is on board with pure-blood wizards being the rulers and ruling over Muggles also, and she’s willing to kill and torture as much as she can toward that reality.
Loyalty to Voldemort is what holds Bellatrix together. I think she worships him so much that he is like a god to her. In the beginning of The Deathly Hallows, when Voldemort is holding a meeting at Malfoy Manor, Voldemort picks up on Lucius’ dismay at Voldemort using his home as Headquarters for Voldemort and he addresses it…
“Why do the Malfoy's look so unhappy with their lot? Is my return, my rise to power, not the very thing they professed to desire for so many years?” “Of course, my Lord,” said Lucius Malfoy. His hand shook as he wiped sweat from his upper lip. “We did desire it — we do.” To Malfoy’s left, his wife made an odd, stiff nod, here eyes averted from Voldemort and the snake. To his right, his son, Draco, who had been gazing up at the inert body overhead, glanced quickly at Voldemort and away again, terrified to make eye contact. “My Lord,” said a woman halfway down the table, her voice constricted with emotion, “it is an honor to have you here, in our family’s house. There can be no higher pleasure.” She sat beside her sister, as unlike her in looks, with her dark hair and heavily lidded eyes, as she was in bearing and demeanor; where Narcissa sat rigid and impassive, Bellatrix leaned toward Voldemort, for mere words could not demonstrate her longing for closeness. “No higher pleasure,” repeated Voldemort, his head tilted a little to one side as he considered Bellatrix. “That means a great deal, Bellatrix, coming from you.” Her face flowed with color; her eyes welled with tears of delight. “My Lord knows I speak nothing but the truth!”
Loyalty. Bellatrix has ultimate loyalty for Voldemort. By killing Dumbledore, Snape effectively dismantled any lingering doubts in Bellatrix’s, Voldemort’s, or any Death Eater’s minds. By killing Dumbledore, Snape solidified his position within Voldemort’s ranks. Even Bellatrix was fooled.
Snape died before he witnessed the victory of the good side. He didn’t get to see all he labored for, but through his efforts, Snape can claim some of the victories the good had over the bad in The Deathly Hallows.