Is $3,556 to install flooring, shoemoulding, and stairs a fair price?

I am getting the entire third floor replaced with Engineered Hardwood. I will be providing the Engineered Hardwood material myself, and the contractor will provide installation and other materials. I spoke to 4 contractors, and this local contractor's quote is the best one so far. I am confused if I should replace the carpet flooring of stairs with engineered hardwood, considering we're going to have a baby soon (cushion safety) compared to engineered hardwood, but it may not be aesthetically pleasing, and babies could potentially chew the carpet. I'm leaning towards keeping the carpet on the stairs for now and replacing it later. This is a brand new build, a Quick Move-in home, so I didn't get the opportunity to go through the design studio. The builder gave carpet on the top floor where we have bedrooms and Engineered Hardwood elsewhere. I need to double-check with the contractor on the materials I need to buy for this job, specifically glue and stair nosings. I'm trying to get a couple more estimates to see if there's a pattern, but both individual contractors came around the same price compared to big box stores. The contractor seems reputable with good reviews and is bonded and insured. The project is expected to take two days.