Seems on you have one foot on the ship and on one the shore! But the good news is you're almost there!
I suggest leaving your shared host. If they are unable to support you with such a request, it's a good sign you've outgrown the service. There are a few options for you to consider.
First of all, when moving web hosting from shared to AWS, you also need to consider what to do with the DNS, email and cPanel services. I use AWS for web hosting, but separate providers for DNS and email. I don't use cPanel - I just configure DNS and the web server manually. This keeps things much simpler and much more flexible, and the only extra cost is a bit of time to configure DNS separately. cPanel, Plesk and similar systems add a lot of unnecessary complexity into Apache and I find this causes problems later.
Though, if you want to keep cPanel, you might consider installing it or some other web-based management system on AWS. (I'd bet you'd find a prebuilt AMI for this if you look around.)
I'm not sure about running a DNS server on AWS, but I think it would be much easier and more reliable to use a DNS service.
EasyDNS.com and No-IP.com are both great DNS hosts - I've used No-IP for my enterprise AWS web hosting for over 2 years. (It is particularly good because they offer monitoring, and automatic and manual DNS failover in case there's a problem. But, that may be more than you need.) I've used EasyDNS for 4 or 5 years. Both services have solid support and are very reliable.
If you want something free, MyDomain.com has been very reliable for me for almost 10 years, but support is very slow. MyDomain will host your DNS for free even if you didn't register the domain with them.
One last consideration in addition to these: Amazon also offers Elastic IPs, which is basically a static IP for your web server instance. Using this will make your DNS much simpler, give you flexibility to easily change to a new instance in the future, if you ever need to. I strongly recommend using an Elastic IP.