I don't know much about dogs and carts. I would guess, though, that it simply feels completely and totally unnatural to them. I would think it would take a LOT of getting used to, especially since, unlike humans, they don't comprehend why they have to use it and the future benefits it will incur.

I would suggest making it a postitive experience for her every time she goes in it. If she like alone time with you, then let that be her reward for spending fifteen minutes in it. Use treats, attention, whatever she loves. Make her associate the cart with getting petting, or getting a bite of liver, or whatever trips her trigger.

I, also, did not find the video to be that sad. Of course, she's your dog and you know her best. But, while she seemed a bit puzzled and seemed to be saying, "Ok, enough already, are we done?", she didn't seem terribly stressed or upset to me. My dogs give me that same look whenever I brush them, take their picture, make them sit and stay.

That little cavalier is precious, giving her kisses over and over. But, perhaps she feels a bit threatened by the other dogs? Unable to get away?