Unless you are designing a intricate one of kind master piece, incorporating multiple artists there’s no chance you need an architect.
The designer and the contractor are good enough,even if your building jurisdiction wants plans. Some homeowners even DIY these plan requirements. A inexpensive drafts person can do plans if the contractor doesn’t want to. If no plan submittal is required there is absolutely no need for anyone extra.
There is no reason you should be wondering much about the final price. The only items of concern for running up a bill are if the existing plumbing is suitable to cut into and if the existing concrete is good. Also, maybe if the inspector requires something not already scoped out and understood to be part of the project. That shouldn’t happen.
You don’t even need a designer if the bathroom is going to be put together with retail products. The sizes of vanities, toilets tubs, ect. can be gotten from catalogues that the stores will give you. The layout has to conform to a few minimum code standards that any contractor should know.
Consider putting insulation in all walls opened for the project including interior walls. It does dampen noise. Cheap sound proofing under the drywall, along with the insulation,is even better.