I wouldn't worry about a tabelspoon full of shredded carrot. The amount of sugar in carrots is greatly exagerated anyway.
Yes, they are in the high range FOR VEGETABLES, they are certainly higher in sugar than lettuce, but the sugar content is still much LOWER than in any of the fruits or other treats people give to their rabbits.
And since this rabbit doesn't eat any treats, a spoonfull of carrot certainly won't hurt. Once she gets used to other greens, you can reserve the carrot for healthy treats, but even then, an inch or two of carrot every day won't hurt.
While you are slowly introducing her to greens, remember that it may take time. Just keep offering, and eventually the bunny will try to eat them. Hazel often has to be offered something at least 3 times, before she'll even try to nibble on it. The first times she just sniffs it.