Take linen off the list entirely. By the 1840s, "cotton is king" and the cheap cotton prints out of the New England mill towns are finding supremacy over linen for women's dresses.
Since you're working on a dress for a "young lady at home"--not doing heavy heavy labor, because a middle working class household will tend to hire in some help for the truly grimy stuff--you could absolutely go with a light shirting for a nice washable summer dress for puttering about the house and garden and trotting to the neighbors, and the blue and white will feel crisp and tidy.
The deep blue print would also be fine, though the weight and dark colors may be problematic in the summer heat, if you're prone to heat issues. That'll be more of a quilting weight, so heavier and warmer than a summer shirting. But, it's also a premium fabric, so the overall quality of goods is going to be nicer than what you'd find at JoAnns or Waldemort.
There's something about this... it's a very simple "neat" style of print, white and grey, in a shirting... good for a sedate young lady, and neutral enough to be paired with charming accessories for casual visiting: https://www.fabric.com/buy/0464417/windham-shirt-tie-stripe-grey-
This is a very small stripe, and with dress lengths, you'd be under $9 a yard https://www.fabric.com/buy/0441014/kaufman-sevenberry-petite-basics-mini-stripe-blue
How much dirty/grimy labor do you anticipate? If it's just everyday household stuff, you may not have to worry about all grime-hiding, like you would if you were a lower working class household. Middle working, upper working, and into the professional classes do things like hiring out their laundry, and hiring in day labor for heavy tasks as needed; uppermost working class and professional classes may even have someone in daily to handle heavier tasks (even if no one lives in.)