Can't connect our Lightsail Wordpress-Site per PHP with the RDS Database. (Only Timeouts or Failed Connection Attempts) Any Ideas?

It is a timeout error, so there may be a problem around the RDS security group or VPC configuration.
For example, have you already set up VPC peering using the method in the following document from Lightsail?
Also, is the VPC where RDS is running the default VPC?
Lightsail's VPC peering can only connect to the default VPC.
Also, what are your RDS security group settings?
Are connections allowed from Lightsail's private IP address?
https://lightsail.aws.amazon.com/ls/docs/en_us/articles/lightsail-how-to-set-up-vpc-peering-with-aws-resources

Thanks for getting back to us so fast!

I think we checked all your suggestions and got a bit further. The problem remains though: we cant connect to the MSSQL Database over PHP. We think it might be a driver issue and that the connection terminates because of it.

At least its a different error now - 500 Internal Server Error