The enfield is not just right hand side, but upside down. Nevertheless it is quite easy to get used to. I have an XT and an Enfield which means swapping between the two types regularly and it's not a problem. I certainly wouldn't let it influence your choice of bike.
You said you'd like a pre '72 bike as they are tax exempt. Is saving £40 a year really that important?
As 'old' style bikes go the Enfield is a good choice as there is a HUGE amount of tuning mods available, and they can be tweaked towards being not too bad off-road. Hitchcocks do trials conversions that include small tanks, high exhausts, 21" front wheels...the list goes on. You can also pick them up quite cheap. The down side is that they have fairly poor performance by modern standards. The 350 Classic is an 18bhp engine and cruises at about 55mph. Tops you are looking at 65mph unless you are riding down hill with your moustache sucked in. The 500 doesn't have a great deal more poke, so I understand. The Enfield also needs looking after, it's not like a modern jap bike that can go forever on a sympathetic smile, you need to check and adjust and tweak stuff all the time. If you don't like getting your hands oily don't buy an enfield. If you do, they are superb tinkering bikes!
The Triumph Bonnie looks like a nice alround bike which could cope with a bit of green laning etc. There is some chat elsewhere on the HUBB about using them as adv-tourers and the general impression is that they are pretty good. You'd be getting modern performance/reliability with that too.
Another alternative might be a BMW R80 (the one sans fairing). Quite capable off-road apparently and good, solid reliable bikes I understand. (A mate has one).

Matt

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*Disclaimer* - I am not saying my bike is better than your bike. I am not saying my way is better than your way. I am not mocking your religion/politics/other belief system. When reading my post imagine me sitting behind a frothing pint of ale, smiling and offering you a bag of peanuts. This is the sentiment in which my post is made. Please accept it as such!