Modern Vespa

So I had the first incident on my 2013 BV350 last week. After work, I rode downtown to get a coffee and when I returned, the bike would turn over but would INSTANTLY die with throttle. I did as much road-side diagnositcs as I could, but never got it running.

Got a tow home and now have some questions.

Reading through a bunch of threads here gives a lot of advice. Battery, spark plug cap and evap system seemed to be the top three. I will double-check the battery, but it was performing fine leading up to the incident. Spark plug cap (and new plug) on order from ScooterWest, to be installed on Friday). I would like to "remove" the evap system from the equation, as well.

When I had a BV200, the "quick and dirty removal" was to disconnect the hose at the carb. I did this after the first signs of symptoms and rode another 20k miles with no issues.

My question: If I wanted to do a similar fix to the 350 without actually removing the entire system, which hoses would I cut, plug and filter? Ideally, I'd like to do it all under the seat until I can rule the evap system in or out as the culprit.

You are headed down one hell of a rabbit hole. We need a better history of your bike to help. My 2014 BV needed an ECU remap at 6000 miles. 2013 was the first year of production, so the flaws needed to be solved.
I bought 2 plug caps from SW and due to poor packaging for shipping, the first cap I installed failed immediately. Now I get my caps and plugs directly from NGK.
If you're going to disconnect the evap, you might as well remove the whole thing and enjoy the extra space.
If you don't know the battery's history, replace it.
My plan of attack would be. Evap, plug cap, battery and if that doesn't get it, remap the ECU(at your nearest dealer) unless it's been done.
Good luck.

Total removal will be a future project. I just can't tackle everything involved in removing panels, etc. I just need a confirmation on which tube to plug and which to filter: Manifold and Overflow.

My nearest dealer is 50 miles away, so that'll have to wait until I get around to buying new tires.

Here is the link to the evap removal thread. This will answer your question. When you are ready to remove the whole deal, the last two posts describe short cuts that I wish I had thought of when I removed mine.

Removal of Evap Canister on BV 350: how-to guide

For expediency, I slightly re-arranged your suggested order of troubleshooting: Spark plug cap (and plug), battery and (not yet done) evap.

I had the plug parts on order, so I was eager to get them installed when they arrived Friday. With a little fiddling (the space is pretty tight to get the plug cap screwed on in the given space with the limited length of cable), I got the new NGK cap installed with a brand new plug. Sure enough, it started up and remained idling with plenty of throttle! Does this mean I solved the problem or that the evap canister de-saturated enough to allow me to THINK I've solved the problem?

I pulled out the battery to find a sticker indicating it was installed in 2014. While the voltage reading from the bike never indicated it was too low, I decided to get a new one.

A rather too short test ride went off without a hitch. With vacation next week, I plan to do a full evapendectomy.

When I get new tires at the end of summer, I'll have the shop flash the latest firmware, if needed.

You don't have to remove the side panel to get the bracket out.
Just loosen the screws mounting the bracket with some pliers. Then spin the screws out as far as you can with your fingers, then use some needle nose pliers the get behind the bracket to get the screws the rest of the way out. Cut one line, plug the other and you are done.