Use VPN connection in certain programs only

I shall set up an open-source software on a linux Server that is in a network which is otherwise not connected to the internet. I have to connect my Windows PC to that network via VPN (type-in a one-time password received by cell phone, start PuTTY, paste SSH private key password, sudo). The VPN is provided by Cisco Anyconnect. As soon as I connected, I cannot use the internet from my local PC any more either. Whenever I need a file from my network drives or form a public source (GitHub, etc.) or need to google for something, I have to disconnect everything, then connect everything again. After every some connect-disconnect circles, my Windows’ network stack gets confused and I have to reboot my PC to get internet connection again at all.

What I would want to have is a local configuration where PuTTY and WinSCP use the VPN connection, meanwhile any other programs (first of all, Firefox, but happy also for Windows’ network drives, Outlook mail etc.) do not.

I have learned that I cannot distinguish this on the “port” level because VPN works on a low OSI layer.

I tried with ForceBindIP and wired network card and wireless network, but I cannot get Firefox to even Google. Maybe Anyconnect intercepts all traffic to both network adapters, maybe Firefox is bound to the network adapter with internet but uses an DNS service which is not, I have no idea.

Any suggestions how to get a workable setup or other workable approach woud be appreciated.