Ezekiel 44:2 would be seriously compromised.
Nope.
Read 45:22 and 46:16. Keeps things in context. The prince is offering a sacrifice and has sons. This is not Messiah.
45:22 and 46:16 have moved on to a different topic from 44:2
Ezekiel is prophesying about the messianic age in Chapters 40 -48. The fact that the prince is not the messiah does not negate interpreting the Ezekiel 44:2 permanent gate closure as a prefiguration of the perpetual virginity of the Blessed Mother who also contained the Lord.
Two different matters are being discussed. Because the prince is not considered to be the Messiah changes nothing about 44:2..