A possibility is that the Golden Horde makes the kingdom a vassal state,and the Horde stays stronger,this would probably require no Black Death and no Tamerlane.
The state could probably resist till the 16-17th century, i do not know how to make it survive till the present day.
I think that a surviving and fairly strong Byzantine Empire which controls the Black Sea would be a better bet.
During the Comnenian period, the Byzantines
did control several cities in the southern Crimea, and the Gothic principality of Theodoros was a vassal of the Byzantines. But after the 4th Crusade, the Byzantine territories in the Crimea came under Trebizondian influence, and the Empire of Trebizond was too weak to make a difference.
But a strong Comnenian Byzantium would have what it takes to hold on to the Crimea, and since the Crimean Goths were Orthodox, the Byzantines would also have a good reason to support the Crimean Goths againest the pagan (and later Muslim, provided that the religious developments remain similar to OTL) Qipchak and Mongols.
All you need then, is to throw in a strong and ambitious Crimean Gothic prince or king who has the support of the Byzantines, and then make sure that the Crimean Gothic kingdom conquers the central and northern regions of the Crimea, perhaps along with some coastal cities/city states near the Crimean Peninsula.
Then the Byzantines would have a strong ally/vassal in the north, and the Crimean Gothic kingdom could very well become a buffer state between Byzantium and the Russians later on.