Firstly, you may wish to check your contracts with UK and US ISPs. You may find that, even if you have an "unlimited bandwidth" service at both ends, the ISPs may have what is called a "Fair Usage Policy" whereby if they consider your usage to be excessive they can break/slowdown your connection (it's done automatically by some ISP's). Note: in the UK your videostream uses the "upstream" bandwidth, and in the US "downstream" bandwidth.
Back to your main question...
How can I connect once and forget about it?
In my experience, the networking side of the SlingCatcher is not designed very robustly. The firmware does not appear to rigorously retry broken connections. It would not be beyond the wit of man to program the firmware to attempt a new connection every 10 seconds or so if the connection dies - but I've certainly seen no evidence of this.
I believe that 2012 will see internet video broadcasting by traditional TV Stations go international. Once this happens, it should remove the need for many of us to rely upon the ill-maintained SlingCatcher. It may indeed be because of this imminent international broadcasting that Sling has decided to no longer maintain the SlingCatcher?
I doubt if you'll see most of the, even if paid for, British stuff streamed internationally through the net, at least enough to justify doing away with the slingbox and slingcatcher. Even using a VPN, free streaming services like TVcatchup ( mostly for the freeview service) still block the VPN IP address, so watching bbc's iplayer, etc and sky news streaming using a vpn are a fall back position to let you only watch the basics, but the genius for me is a Virgin V+ box DVR connected to my slingbox allowing me, because of the time difference, to record everything and then watch it on my slingcatcher in the U.S. I think the best we can hope for to keep the sling philosophy intact is for slingmedia's software to be provided with google tv or any stand alone box or tv that could use sling's software and stream that content from the slingbox. Surely, something like that has to be in the works.
As for the original poster wanting to leave things on 24 /7 on both the slingbox and the slingcatcher end, I really wouldn't recommend it. Constant use and heat could easily cause either the power adapter or hardware to fail. The solo is already notorious for cap blowing and I believe that's due to heat. No reason to leave them on all the time and put stress on the components when a minute is all it takes to connect the slingcatcher up to the slingbox. That's my philosophy that heat is the enemy of electronics and so far, touch wood, everything has been working smooth for me on both sides of the pond for years, so I stick to that philosophy.