Having worked for a couple of hosting companies I can tell you that turning off mod_security is not going the help you stream from Bluehost. What they didn't tell you is that you are not going to be able to stream from a shared account, you are looking to host the json file you may need to check out their VPN's, that way you can use the account for what you want. They can unlock anything that you want them to on a VPN account.
I recommend taking a look at their "Acceptable use policy" and their "User Agreement". Both of those indicate that they host websites, not storage for other services of any kind.
If you have a Bluehost VPN or a dedicated server you might be able to do it but there may be bandwidth limitations. I have a vimeo pro account, that is great as a content distribution site, then you might be able to host you JSON files there, but I never had any luck with that in my Bluehost account. I have a couple of dedicated servers on other host companies that host my json files, and I have one set up with another channels worth of my videos too. I use "https://www.server4you.com" it works, and has no bandwidth restrictions , also "https://www.wholesaleinternet.com" but there is a 100TB bandwidth restriction, I haven't tried setting that one up for direct video distribution so I don't know how that one does.