What is the ruling on deliberately going to a specific mosque to combine prayers due to rain, while there is a mosque available that does not combine? Is this considered seeking out concessions (tadayyu' al-rukhas), and is it like the ruling on someone who travels during the day in Ramadan not for the purpose of travel but to break their fast?

This action is similar to someone who travels in Ramadan (definition) to break their fast. Perhaps their effort in searching for another mosque where they could combine prayers would be greater than waiting for the 'Isha prayer.

It is considered that his prayer is closer to invalidity, because he went to avoid the prayer, not for ease. Prayer was prescribed as a timed ordinance, and performing it before its time without a valid legal excuse is not accepted. The advice for someone who does this is to pray 'Isha at home if the rain causes them hardship, and they will receive the full reward of congregational prayer due to their excuse.