How to do automatic 2FA with SSH/SCP on Windows (ideally using PuTTY+WinSCP)?

Neither WinSCP nor PuTTY provide a way to automate the 2FA authentication.

And even if they did, a question is whether the system that provides you the code (which you didn't tell us anything about) would allow you to obtain it automatically. It would kind of defy its purpose.

A possible solution is if your client IP addresses are static, which is to set SSH on the Linux server to skip authorization for your IPs.

The post SSH Google Authenticator Ignore/Whitelist ips has this answer:

In "/etc/pam.d/sshd"

auth [success=1 default=ignore] pam_access.so accessfile=/etc/security/access-local.conf
auth required pam_google_authenticator.so

Then on "/etc/security/access-local.conf"

#localhost doesn't need two step verification
+ : ALL : 192.168.1.0/24
+ : ALL : LOCAL
+ : ALL : YOURIPHERE
#All other hosts need two step verification
- : ALL : ALL      Don't forget to restart ssh.

Beside IP addresses, I don't see another way of distinguishing the clients. You might consider using a jump server having a static IP, so that these clients will connect through it and will this way be exempt from authentication.