I will acknowledge up front that I do not yet have a great deal of experience playing with these cards and therefore may not have fully grasped the "meta." However, I have had considerable success in the past couple of years playing EIK, mostly tournament 4ers, on another site. I was finding play there to become a bit stale in large part due to what I felt was the over-sized importance of one's place in the FG queue, that, as a result tended to dictate stage 1 actions.
I was pleased to find that these new cards mixed things up with a number of "growth queue disruptors" (e.g. cottager or stork) and some alternative early game strategies (e.g. big country).
In this latter category I would place field watchman (plow and sow in stage 1) and clay supports (reno first, then build). For this reason, I would hate to see these cards banned. Field watchman is certainly a strong card, but not game breaking. It also seems that there are now a sufficient number of other cards that support plowing(plow man, plow driver, assistant tiller and 3 or 4 plows).
Clay supports by itself is a good, but not over-powered card. When played in the "traditional" way (reno, then build rooms), the first room actually costs 3w,4c,2r (2w for card, 2c, 1r to reno, and 2c, 1w, 1r for room) and generally can't be built until well into stage 2. There are a number of other cards that give similar room discounts (carpenter, carpenter's parlour, wooden hut extender), that quite reasonably are not being discussed for banning. Yes, certain combos (e.g bricklayer) can make rooms ridiculously cheap - in one of my last games on the other site I lost to a 15 room stone house built with clay supports, bricklayer, and ladder. If combos could be banned, you might get my support, but I don't see banning a single card in order to disrupt a powerful combo as these combos will come together infrequently and, as fiscused suggested, there are many powerful combos in this game.
Regarding the other suggestions, I don't see lantern house as inherently over-powered. It is most analogous to tutor. With tutor you can get 6 points for playing all 7 occs. Lantern house gives you 1 more point for additionally playing all 7 minors. BTW, I had these 2 cards along with education bonus in a recent game. Yes, over-powered combo, but again, I think each of these cards is quite reasonable by itself.
Carriage trip? Annoying, yes, but I kind of like the disruptive potential and by itself, really doesn't do anything else. The fact that it is a passing minor also means that your opponents have an opportunity to return the annoyance. I say keep it.
Begging student? I'm not sure. Does seem powerful. For me, the closest analogy is educator in the original EIK. With begging student you can potentially get out your other 6 occs, skewed toward the end of the game, at a cost of 3 points. With educator, the occs can, and typically do, come out much earlier, and will cumulatively cost 14f. Educator is certainly strong, but not banned. I guess I would take a wait and see approach with begging student.
Also, wondering if the plan is to eventually get the remaining cards from ABCD on this site. If so, I would be careful about banning cards now unless there is experience and precedence from other venues, as there was with braggart and caravan.