Are there better (easier) ways to get a specific domain's SID?

What you're doing looks like the best option to me. Hardcoding strings is definetely not a good idea.

Each domain has a build in account domainName\administrator, so you can create an account with this name, translate it to the SecurityIdentifier and read the AccountDomainSid property.

An example of this way is:

public static class SecurityEx
{
    public static SecurityIdentifier DomainSId
    {
        get
        {               
            var administratorAcount = new NTAccount(GetDomainName(), "administrator");
            var administratorSId = (SecurityIdentifier) administratorAcount.Translate(typeof (SecurityIdentifier));
            return administratorSId.AccountDomainSid;
        }
    }

    internal static string GetDomainName()
    {
        //could be other way to get the domain name through Environment.UserDomainName etc...
        return IPGlobalProperties.GetIPGlobalProperties().DomainName;
    }
}

Also you can find other solutions to achieve the same result via WMI or Lsa. This one seems the most elegant way for me.