That’s a good requirement. You will get a decent machine as a VPS for this money. And it’s actually not that expensive if you know where to look for it. My suggestion will be host-world. You can check the reviews here https://host-world.com/reviews . I’ve been using it for years, and won’t pick another service.
Since you’re looking for relatively budget solutions the custom software is out of the question. Some junior developers may offer you a very lucrative price- but avoid this offer unless you want to receive a non-working mess. But you can always just buy this software https://claimgenix.com/medical-electronic-claim-software/
You don’t have that many options after all. After spending days researching looking for the platforms you will end up only with several options. This one will surely be on the list https://nybillpro.com/ and I strongly recommend you to check it. It has educational materials there as well.