Peasant
...say that I go with your second solution, in that my SME box has the domain mycompany.uk.com, and the main server in the USA (mycompany.com) forwards mail to it.
Firstly the mail server parts of sme still work whether the box is in gateway & server mode or server only mode. Only private server mode disables many external services, and rightly so as it is in "private" mode which means it cannot be seen from the Internet and no ports are open.
OK, if you set your USA server with mydomain.com, then I suggest the following.
You can obtain a perfectly functional and free domain from dyndns.com (and many other free domain name providers) eg mydomain.dyndns.org (or other variants), and give your UK box that domain.
Note that in Australia real au domains cost approx $AUD45 for two years, so that's not a real significant cost for any business, if you prefer to register a "real" domain.
For a "real" domain, you can get free DNS hosting at
www.zoneedit.com, and configure and maintain it yourself, quite easily.
I would forget the drop box idea as it's not really necessary, just use both servers as smtp mail servers, which in most default configurations is already running. If the UK server is turned off overnight, then qmail on the USA server will queue the mail for delivery when it cannot get a connection with the smtp mail server on the UK server. It will keep trying to deliver the messages for 7 days before it finally gives up and sends the original sender an undeliverable message, but of course the UK server will come back on in that time and accept the mail messages (on the next scheduled retry).
Of course it will work.
In server manager you configure the user accounts (for UK users) in the USA box, to deliver all mail to UKuser1@mydomain.dyndns.org (or whatever domain you chose). When mail for UKuser1@mydomain.com arrives at the USA server it automatically gets forwarded to the nominated users address.
As mentioned above if the UK smtp server is offline, then the USA server will queue the messages to retry delivery later.
The domain name mydomain.com exists anywhere in the world (that's what DNS & MX records are for), and so therefore does UKuser@mydomain.com and USAuser@mydomain.com.
The sme server's smtp server will send email to the correct external smtp server that is configured for email for the domain used in the email address.
As sme is easily maintained remotely, then it's just as easy to configure users on the USA server from the UK, as it is to do so locally (& vice versa).