I think according to usual romanization rules, 五兵衛 should be romanized gohee, as you say.
That said, the そば屋 you link to actually provides a reading of its name そば屋 五兵衛 【そばや ごへい】. So they're not romanizing ごへえ, but are romanizing ごへい, which would be gohei. Whether their reading of 五兵衛 is nonsense is a different matter.
I can only guess that whoever tried to romanize 五兵衛 from the other site you linked thought that 五兵衛 should be read ごへい (which isn't hard to imagine since ごへえ is an irregular spelling and both ごへい and ごへえ are pronounced the same).
So, you continue to romanize 五兵衛 as gohee and pride yourself in knowing that the reading of 五兵衛 is ごへえ, with an え.