Given that the Kazakh Dandelion apparently has deep roots, maybe a Roman Patrician notices that fact and decides to grow the Kazakh Dandelion on his estate to make sure barriers in the ground soil are broken up and crops grow easier.
The issue with that is that Italian soil (and soil across much of the Mediterranean), fundamentally, doesn't have that problem. It's relatively thin. That's why Romans generally used scratch plow rather than heavier wheeled plow (though the latter were not unknown).
I say
generally because Pliny (writing in the 1st century BC) remarks that a wheeled plow was in use in Asia Minor, though it seems to have been halfway between the earlier ("ard") and later ("mouldboard") versions--wheeled, but no boards (as later heavy plows would have). Evidently, Roman patricians didn't see it as particularly appealing, since it did not reach Italy until it was brought by the Goths centuries later.
If anything, I'd expect the dandelion to be introduced in this manner if you had a surviving Celtic polity focused on northwestern Europe, rather than a Roman conquest, but this would require a POD at least during the life of Julius Caesar (him dying before conquering the Gauls), if not during the Punic Wars.