Current record is about 1200 km. Is there a physical limitation to this distance?
Depends upon what was tested at the 1200 KM that limit you mentioned. If the photons were termed to be entangled because they were found to be anti correlated (i.e. opposite spin/polarity), then there is no limit of distance. Actually distance is not even a factor in case of anti correlation. Once you create them as anti correlated, they will remain so until disturbed from that state.
So testing and demonstrating just opposite spin/polarity at larger and larger distances is a waste of time because anti correlation alone can easily be explained in terms of hidden variables.
If they demonstrated statistical correlation, in addition to anti correlation at a distance of 1200 KM, then also I do not think why there would be any limit. What would change if it was 1400 KM?
Therefore, there is no distance limit on entanglement. I would say, distance is not even a factor to be considered. FTL can be very well tested at much smaller distances.