Is there something that the BI app doesn't do that you want? It does not require a VPN but you already have that set up now anyway and that's the better way to go.
In any case, looking quickly it appears that TinyCam does support some sort of connection to BI:
How do I use the cam number on Tinycam to select a specific camera or camera group? the channel numbering in Tinycam seems random in terms of which BI cam it shows on Tinycam. Sometimes it shows multi cams in a split screen on Tinycam when I select one of the higher numbers for channel...
ipcamtalk.com
Generally, any app will work in the way that I described above - you'll be a client on your local network when connected via VPN. Not strictly true, there are some differences, but effectively so in most all cases. When using the VPN, you don't truly have a local address. You have an external address which is routed internally over to your local network. The router/firewall evaluates that as an outside address, so the rule blocking access applies. There's a way to pass the TUN (VPN) interface through but it's fairly complicated and requires command line changes to the Asus config.
The problem in connecting to the cams isn't the VPN (or lack thereof). It's that you have access to them blocked. You could turn that off and they would work VPN or not. Better to block/segregate them though.