Never been to Beckley myself but the description you gave of the cold spot shifting location across the room is something I find very interesting from a poltergeist research standpoint. Static cold spots are often dismissed as draughts or thermal gradients but a moving cold spot that tracks independently of air currents is a different matter entirely.
Back in 2019 I documented something similar at a property outside Dundee and we couldn't account for it with any conventional explanation after three nights of thermal logging.
What equipment were you using when you noticed it? And did anyone else in the group feel it at the same time or was it just you? That question of shared vs individual perception matters quite a lot when trying to rule out suggestion and expectation bias.
Would genuinely like to hear more detail from anyone else who's visited that farmhouse. Even negative experiences - people who felt nothing - are useful data points.