Ok- First confession- I've owned none of those models, but I have been studying them for nearly a year, and like you I do own an older Vespa. (I won't insult you by claiming mine's vintage, it's a 1980).
You've mentioned all the scooters I've either looked at or considered, and I'll explain why I haven't bought anything but still feel I can weigh in with some advice (it's free afterall).
I'll cut to the chase- for what you're describing as the designated purpose a nearly new HD200 would be perfect for you. They are reliable, comfortable (can actually seat two up) and you've got a really knowledgable dealer in California- http://www.ootyscooters.com/
. He posts lot of great info on the SYM forum, and is a huge fan of the HD200.
I'm sure you're also aware an HD200 has previously won the cross country Scooter Cannonball.
The SC250 is a great scooter- Klaviator on here has toured on his, and has great things to say about it. Between it and the SYM the bigger issue is whether you have access to a local dealer for any service items you might want addressed.
The BV350 and 300GTS are sort of in a league of their own. Complete apples and oranges compared to the utilitarian HD200, and with prices to match- both are great, but do some research to see if they fit your real philosophy of use. Seems like from your description both are overkill. Again, dealer access is important, and lots of folks will caution you that Piaggio support in the US has been a bit spotty. Spend some time on Modern Vespa and you'll get to hear all the great and not so great things about both. Only downside that I have for the GTS are the passenger seating position is sub optimal due to the wide rear end of the scooter. My wife could barely reach the pegs, and indeed it's a common complaint.
I came close to buying an HD200 to go on the back of my RV. Alas, when I went back to the dealer they had sold the flat grey model I had lusted for, and the red and canary yellow models did not mess well with my RV colors. So for the time being I'm going to use my P200E for carry duty, and perhaps if I sell my R1200GS next spring I'll look for a bigger replacement like the BV350. For now I'm waiting to see how badly the 'fiscal cliff' affects me
Hope you find the perfect scooter for your needs.