hey Waco, I've been pretty busy lately and haven't been on the site much the last couple months but thought I'd take a few minutes and reply to your help request since no one else has. I had a 2009 GTV 250. Great bike.
Those early 250 bikes had a couple common problems. Most older than 2008 GTS/GTV 250s had a problem with the fuel pump. Your running/stopping symptoms are exactly the behavior the original fuel pump problem exhibited. What would happen is the fuel pump would warm up, a part of it would expand and bind the pump so it stopped rotating. Once cooled off, the pump ran fine till it got hot again. The fuel pump is in the fuel tank. Some people claimed keeping the tank full-ish helped keep the pump cool. That makes sense but I never had a problem so I've no direct experience.
I "think" an important diagnostic is to listen for the fuel pump whir. The fuel pump powers on for about 2 seconds when you turn the key to the on position. Do not push the starter button. The whir is soft but is easily audible. Try it with your bike when it's cool. This should happen every time you turn the key to the on position. I remember someone telling me they went on a long trip in hot weather and they ran into this problem. They continued on the ride but had to stop for 35-45 minutes till the fuel pump cooled off before they could continue. I think when the pump gets warm, it cannot start rotating and you won't hear the whir.
Vespa ultimately ran a recall campaign and replaced most of the 250cc fuel pumps free of charge. I see your's was replaced but did they replace it with the new, updated fuel pump. You need to ask the person that had the bike before you. It might be hard to find out. If Vespa did it, any dealer should be able to look in the Vespa database and see if your was upgraded on the recall campaign. If not, it's possible you might have to just replace it. Vespa may still do it for free if it wasn't already done. I don't know. You'll have to ask a dealer.
I did the fuel pump recall on my bike. it took about 75 minutes for a knowledgeable mechanic to do it. I gather it's kind of a PITA to access the fuel tank.
If you search the site for fuel pump failure, you will find lots of traffic on it from about 2008 to 2022. For example:
https://modernvespa.com/forum/post2549062#2549062
The only similar problem I and some others had on the 250 GTS/GTV bikes was the the wire from the ignition coil to the spark plug was a little short and could fall off easily. The bike would not start tho the engine would crank over when the starter was pressed on.It was not an intermittent and temporary failure like the fuel pump. The solution was to replace that cable with one that was a bit longer. I think the dealer just bought some wire from the local auto parts store and made up the cable from scratch. I don't know how to do that but I imagine it's simple.
I hope all that helps! Miguel