I know you have all put your best thoughts to this problem - lots of input so far...
but...
Overdraft makes no sense to me - when she shuts down the air she is not closing all the air, simply closing the primary air inlet. If she had an overdraft going on, then the secondary air would take over, and the fire would continue, and the temps would rise, not fall and the fire die out. That secondary air won't just rush up the flue - if the fire is hot, then the gasses will ignite due to the increase O2.
In the end, the fire is not hot.
A block off plate and insulation will not decrease draft, it will increase draft - so It's not going to cure an overdraft - but again I don't think that is the problem. As well, adding a flue damper will negate the EPA design of the stove, and assumes that overdraft is the issue, which is not the situation SherryAnne is decribing.
My guess here is still poor draft due to incomplete burn - i.e. the fire is not hot. There could be a mechanical block with the air inlets in the stove (still a draft issue, as for the chimney to suck air thru the stove, the air needs a way in). As I read the manual for that stove, it doesn't sound like they expect you to get much heat in the "low burn setting (air half closed)" as they don't want you to use the blower in that air setting, so I wouldn't worry about turning the air down early - your flue set up may need a lot of heat to maintain good draft that will allow it to burn on secondary air alone, so don't rush.
SherryAnne - Have you tried my fire lighting technique below? In the end, even if all the convective heat is being lost up the flue because of no block off plate, the surface of the stove should still be getting hot - as in you should feel the heat on your hands when you are running it hot. It wouldn't matter if the unit was outside and it was minus 10*F, the stove would still eventually get hot.
If you truly want to know, then dry some wood in your kitchen stove - after supper, when the oven is cooling off (about 250F and cooling), stick some small splits in there and leave them for a few hours. Then build up a large "log cabin" of splits and kindling, light it, and once it seems ready to go with the door open, then let the stove run with the door closed and the air wide open. Don't close the air until your thermometer shows the stove heating up. then, close it really slow, as I think your flue needs a lot of heat to maintain draft.
It may take 2 full loads burning small splits to heat up your flue looks like a massive stone unit, so it will soak up a lot of heat from your liner, and take longer for your liner to heat up and draft properly.
Do not be in a hurry to turn the air down - when I first installed my unit, it often took 2 loads of fine splits to get the flue warm enough to create enough draft to allow me to load larger splits and turn the air down.
Here is the fire lighting link -
https://www.hearth.com/talk/threads/31393/
If you find that the longer hot fire works, then insulating the liner will help, as the liner won't radiate as much heat into the surrounding brick/stone, so it will get hotter sooner and draft better.
But a fire that dies when air is closed is the exact opposite of overdraft.