VPNs are fine for almost every application with the exception of this. Now, you could do some things on each end to eliminate the broadcasts and reduce keepalives etc, however it just requires tinkering and lots of it.
Everyone has their favorite "combo boards" but really you have 50% of what you need already. Just get a TL-WR802N or something and away you go. The 3G/4G modem you have already has routing built in and will power on with USB connection which is key. It's also a stable unit for this application.
You won't be able to port forward, which is why I said this gets tricky. It's again based on how cell companies hand out IP addresses... they aren't IP addresses that can be visible to the 'outside' despite looking like they're in the right class. Your 3G/4G router will have port forwarding options in its admin menu but don't get your hopes up
You won't be able to connect this to a remote BI server in any productive way to see live stream. There are two methods:
1.) Rely on the cam's "cloud" connection. It's chatty, and I've personally not had much experience with them, but it is an option for some cameras. The streams do not initiate until you log into a website and queue up the camera. You'd have to negotiate bitrates on the camera or look at sub-stream in order to keep your connection happy. Too much data means a significant live lag over cell.
2.) BI server sits locally. I don't bring this up often to other techs because they start swarming me with "BUT WHAT ABOUT--" and "YOU CAN'T DO THIS" comments, but after many successful installs, I don't care. It works. And can work well with solar. Trick is finding a PC that runs on 2-5W with BI running. Which they exist. The Windows based pc will allow you to set up a non-chatty VPN proxy and away you go. You can then bypass the camera's motion detection, FTP, e-mail, etc settings and also allow you to do all the fun things you want. The major trick here is ALL OF THE FACTORS that you have to tune and be aware of that will give you headaches. If you have the time to do all of that, it's worth it.
Here's what my installs look like size wise to give you an idea of what would sit on the tower (minus batteries and panel, of course
)
https://www.ipcamtalk.com/showthread.php/12780-4G-LTE-setup