Since most people in the UK spell it without the accent (Amélie) and pronounce Amelie to rhyme with family it doesn’t sound very posh to me.
Most of the people I consider well bred and sophisticated chose straightforward names, often with a family connection, like Catherine, Charlotte, Alice, Caroline, Elise, Lucy, Julia, Elizabeth, Sarah, Rosie and Sophie, Amelia, Chloe and Isabel/Isabelle. Some of the more creative names in my well-heeled village are Violet, Xanthe, Zara, Tamsin, Edwina, Lucinda, Theodora, Clara and Bronte.
Posh is as posh does - it’s not the name that makes a baby posh, it’s the family. So if your husband doesn’t feel posh, it won’t matter what you name the baby.