I use gl.inet routers on both devices, but other brands probably do ok as well. The VPN software is pre-installed on their devices. I prefer using Wireguard however both OpenVPN and Wireguard are there. On the routers, you create public and private keys but you only share the public keys between them. You can also create a QR code public key which you can scan with your phone or laptop. Use the standard Wireguard app for your phone. The default key setup will make your clients appear to be in the LAN at home whenever you enable the VPN.
Maybe they can see that you're using an IP from a known VPN provider. I run VPN on both my home router in US and home router in Thailand and never seem to have any problems. I have seen a few sites block Surfshark VPN
I did exactly what you are asking about two years ago. The US embassy included a letter for Thai immigration requesting them to transfer the visa into my passport. After that I was successfully able to apply for non-O at Phuket immigration