If your bike is having the same problem, dying after a mile, with the Petcock set on the "prime" position, I wouldn't think the vacuum diaphragm is the problem.
You might still have an issue with fuel supply though. Test by disconnecting the fuel line from the carburetor and let fuel run into a pickle jar with the petcock set to "prime". I don't know the design rate, but I'd expect something like a half glassful after a minute, minimum.
If the flow is restricted, it could be a few things.
- clogged fuel filter
- obstruction / kink in the fuel lines
- clogging at the screen above the petcock. Remove the petcock to access this.
- clog inside the petcock
When the engine stops, there's two tests to do.
- Open the fuel tank, listen for it "taking a breath". If the tank sucks in air, and the issue is fixed, that suggests the tank vent line is blocked. If the vent line is blocked, the issue occurs faster with a FULL tank than with one almost empty.
- If the tank doesn't suck air, open the bowl drain screw at the bottom of the carburetor and see how much fuel drains out through the drain nipple into a small container. Compare that amount with how much drained out when you did the same test while the engine was running fine.
https://i.ebayimg.com/images/g/R1sAA...i/s-l1600.webp
My GZ250 does the same thing you describe with the carb being slow to react, especially when the engine is hot. I'm not sure if this is a characteristic of the Mikuni BSR32 on this engine, but I think it is typical of a CV carb on a lean running engine. My machine is running quite lean with the factory jets and equipment. I like the fuel economy, so I haven't changed it. Another thing that would cause or exacerbate this behavior is an air leak in the intake boot, or the vacuum system.
My choke lever will hold the "On" position itself.
I use one of these inexpensive inductive sensing tachometers for setting idle and general testing. With four or five wraps around a spark plug wire, these attach easily and are accurate at lower rpms.
https://www.ebay.com/itm/162383022438
I don't generally ride around with it. The battery has lasted for several years, mainly just sitting around. The battery isn't supposed to be replaceable, but when it dies, I'll crack the case open and replace it anyways.