Would the opinion of Bellatrix Lestrange have changed if she knew Lord Voldemort was a Half-Blood himself?
Bellatrix did know and her opinion did change.
“You dare speak his name?” whispered Bellatrix. “Yeah,” said Harry... “Yeah, I’ve got no problem saying Vol —” “Shut your mouth!” Bellatrix shrieked. “You dare speak his name with your unworthy lips, you dare besmirch it with your half-blood’s tongue, you dare —” “Did you know he’s a half-blood too?” said Harry... …“Voldemort? Yeah, his mother was a witch but his dad was a Muggle — or has he been telling you lot he’s pureblood?” Harry Potter and the Order of the Phoenix – J.K. Rowling
Not long after learning the Dark Lord was a hypocrite, Bellatrix began to have her doubts about Voldemort’s decision making.
Bellatrix was obsessed by the power Voldemort held; she followed him blindly for over twenty years, including nearly all of Harry's life in prison because of the Dark Lord.
Apparently, an extra fifteen years inside the walls of Azkaban wasn't quite enough time to turn Bellatrix against Voldemort.
But Harry’s declaration seems to have been the feather that broke the hippogriff’s back.
Shortly after the Battle of the Department of Mysteries, Bellatrix would disclose to her sister, Narcissa Malfoy, that she believed Voldemort wrong to trust Severus Snape.
But, wizards and witches are as human as any muggle, and Bellatrix was emotionally fixated on the former Tom Riddle; some have suggested she was in love.
She remained loyal to him until the end; it seems even Bellatrix wasn’t as completely convinced about the need to keep wizards pure and separate from those wretched Mud-Bloods