cURL is already installed at your server; there is no need to install it as it is already available.
In order to use cURL with your program, your program will have to be modified. While you have not told us what program you are using, you did say that you had "bought" it, so I suspect you may not be able to share the code with us, and you may even be prohibited by its license from modifying it.
If this code is not something that you can legally "share" with others, this makes it impossible for us to evaluate how difficult it would be to re-write it to use cURL instead of allow_url_fopen, or to actually modify the code if it was something someone here wanted to do.
Again, you should contact the author of the script and request a version that does not rely upon the availability of allow_url_fopen. When you buy proprietary code, without the ability to share it with others , you really have no other alternatives if you are not able to modify it yourself or modify your environment to facilitate running the code.
In many cases, you may not legally modify such code anyway, which leaves you with only the alternative of running in the environment the author supports.
Unfortunately, in this case, the DreamHost environment does not facilitate you running that program without modifying your PHP environment. It is possible to modify your environment to facilitate the use of allow_url_fopen, but, I'm sorry, I won't help you do that as I feel that it is unwise to allow_url_fopen on a shared server and I support DreamHost's having disabled it.
Edited to add this note:
This is the very same situation the you encountered back in August, with one important difference. In that case you were using "free" code that could be legally shared and modified, and we were able to provide a cURL based "fix" so you could run the program safely. In this case, you have purchased a program that, I presume, does not have a "free" license, and that prevents us from attempting the same fix this time around.