Apple Developer Program Refund Help
I think you have some things mixed up:

The developer agreement is between Apple and the seller of the app (free apps are also regarded as being sold on the App Store). It is not between Apple and whomever actually makes the app.

If you're an individual making and selling an app, you need an "Individual" account with Apple. The name of the seller on the App Store is then your given name. The name of the app itself can be anything - it doesn't have to be your name.

If you're working for a corporation developing an app to go on the App Store, the corporation needs a "Company" account with Apple. This agreement can be made by anyone authorized to sign the documents for the company - for smaller companies this is usually the director. This also applies if you're both the developer, owner and director of the company.

If you're developing an app for a client and they're going to release it on the App Store - then that client needs to enter the agreement with Apple. Either as an Individual or as a Company depending on who your client is. You yourself needs no agreement with Apple at all.

In your specific case, the app is not going to be sold by you - rather it is going to be sold by your client - the hospital. Therefore the hospital needs to apply and enter into the Apple Developer Program. When they have done that, they can create a user account for you with developer access, so that you can upload the app for approval by Apple.

The specific part of section 5.2.1 that you're in violation of is:


  Apps should be submitted by the person or legal entity that owns or has licensed the intellectual property and other relevant rights and is responsible for offering any services provided by the app.


Therefore it does not matter that you state that you have agreements with the hospital that you are allowed to use their name, trademark, logo, etc. The app needs to be submitted by the entity providing the service - not by the person that happened to be doing the actual coding.

You can make a second plea to Apple stating that you didn't know that you weren't supposed to enter an agreement at all - but if they already explained to you that it is non-refundable, there's really nothing you can do. It is explained up front by Apple who needs to enter in an agreement and pay the fees, so it was your responsibility to figure that out before ordering and paying.