It seems like it could easily be done and should effect the structure of the dress since it has the mesh. I disagree that the dress would be less attractive, but I don’t think it necessarily adds to the dress since it’s likely to be covered by your arms most of the day. But if you want it, go for it! I’m sure your seamstress will let you know if it’s not possible. Beautiful dress by the way, simple, classic, elegant!
Disclaimer - I'm not a professional seamstress, but have done quite a bit of home sewing in my day. Can it be done? With the right seamstress, yes....
But in my estimation it is going to be costly. Side seams help determine the structure and fit of the dress, and how it lays on the body. You have a lining in that dress that will need to be separated and altered along with the outer fabric and then all put back together to appear flat and flawless.
I think it would be cheaper to buy the dress that was designed and patterned with that deep armhole.