Does school name matter for me getting a Finance degree?

I am located in Arizona and I’ve found ASU’s W.P. Carey is a pretty good business school. I am currently in an entry level leadership role in municipal government, and have about 6 years of progressive leadership in private and public sector along with sales and administrative skills.

I am finishing my associates and am trying to find the right path forward. Through my workplace, I get access to free education from a smaller private university with a step below in accreditation but I could double major in finance and accounting. I am worried though that having a degree from a more respected school like W.P Carey would be helpful or am I simply overthinking?

At ASU I will need to pay a few thousand out of pocket after scholarships and through my workplace it’s free.

What would you choose?