problem with online class
I can understand your embarrassment, but you may have to overcome it and just ask for an accommodation. A reasonable person should agree.
But many colleges and universities have some sort of office of student assistance for making such requests and passing them on. If that is available it would free you from having to discuss your situation over and over. See if that is a possibility.
And, for what it is worth, I find such requirements to be offensive. Instructors need to find a better way to evaluate students than to intrude on their homes and families. Few students have the capability of setting up a secure testing space and finding others to manage their children, etc.