There are many excellent content management systems you can use, and you would be able to customize them.
However, if you have no experience with the system, 3 weeks is not enough time to deliver a quality solution.
I have been in the same situation you are in - and I did not finish when the client wanted it done.
The goal of any project is to deliver to the client what they want, in a timely and cost-effective manner, and make a fair profit.
If you agree to do the work but don't allow enough time, the project will fail in one or more ways - it will be late, too expensive, or not well done.
You said you like Joomla. Load it onto your server and learn everything you can about it. Spend time on the Joomla site and learn about the available plug-ins and learn from the forums. Check for themes/designs that would work for your project. Try out what you learn to address the client's issues and decide if you think you can do the project in 3 weeks.
Don't allow anyone to create email addresses on your server. It just isn't worth it.
Good luck. The first project is the most difficult, after that, each gets easier and ultimately, it becomes fun.
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