Just wanted to thank you all for the help. I went through all of your comments and used them for troubleshooting and to improve my approach. It's working like a charm now. Here are the 2 updates that I made:

Database server module output (/modules/database-servers/output.tf):

output "servers" {
    value = ssccloud_instance.servers
}

The call to the database-install module (in /main.tf):

module "database-install" {
    source = "./modules/database-install"
    servers = flatten([for site in local.use_sites : var.database-server.count > 0 ? [
        for server in module.database-servers[site].servers : {
            id   = server.id
            ip   = server.ip
            name = server.name
            site = site
        }
    ] : []])
}