Is entering the mosque and exiting from the other side to reach the ablution area considered taking mosques as thoroughfares, given the possibility of reaching the ablution area via a longer route?

The Prophet, peace and blessings be upon him, forbade taking mosques as pathways, except for remembrance [of Allah] or prayer. The meaning of this is making them a passageway without prayer. Passing through a mosque without prayer is disliked (makrooh) unless there is an excuse or a need, such as difficulty in taking another route. This detailed ruling is supported by hadiths and narrations. If taking another route involves difficulty, then passing through the mosque is permissible. If it is for the purpose of performing ablution and returning for prayer, then there is no harm in that.