If marriage is governed by law, then there was no valid marriage and the child of Crown Prince Carol of Romania and Zizi Lambrino was illegitimate. Crown Prince Carol was barred by the Romanian constitution from marrying a Romanian, or a commoner, or without the monarch's consent. He did all three things, a Romanian court ruled the marriage invalid, and while I don't know the exact wording of the provisions I see no grounds for questioning that decision.
It may be that, unlike Britain's Royal Marriages Act which provides that no lawful marriage is possible for those within its remit unless the sovereign consents, their effect was that a marriage contravening constitutional requirements while still a marriage removed Crown Prince Carol's succession rights and royal status, therefore preventing his children from having same. Another court later overruling the original judgement and deciding that the marriage was good after all would merely mean that, instead of Carol Lambrino being barred from succession due to illegitimacy, he would be barred due to his parent's marriage contravening the requirements of the constitution.
Next question, was Crown Prince Carol's second marriage bigamous and King Michael illegitimate? The younger Carol was conceived nearly a year after the annulment and the forcible separation of the elder Carol and Zizi. What happened was that after his release from confinement (as an army deserter who had abandoned his post in order to flee with Zizi and get married -- anyone else would have been shot) Crown Prince Carol took up with someone even more unsuitable, a milliner's daughter of apparently boundless ambition. In order to break up this liaison, his parents reluctantly allowed him to reunite with Zizi. Their son was registered under his mother's maiden name, a clear sign that his parents considered him illegitimate and themselves not wed, which was actually the position in Romanian law at the time.
So when Crown Prince Carol married Princess Helen of Greece and Denmark, there is no reason for questioning that he in good faith believed he was free to marry. In Romanian law he was, and while I again don't know the exact position of the Orthodox Church I imagine it is similar to the Roman Catholic position that the children of a marriage made in good faith are legitimate even if the marriage is subsequently discovered not to be valid. So Carol Lambrino had no succession rights whether his parents' marriage was good or not, and King Michael did, being legitimate whether his
parents' marriage was good or not.
The position of the Church on the question is relevant in that the grounds originally adduced to a court in Lisbon, and upheld by another in Paris then several in Bucharest, are that the marriage was in an Orthodox religious ceremony and never annulled by the Orthodox Church. Another possible ground might be that the ceremony was in Russia (albeit a German-occupied part of it) so not subject to Romanian law, though there was no mention of this in what I have seen.
For the first, it is a strange position for a court to hold that the laws of the Church are above the laws of the state; it is as good as saying that the Church can legislate to overrule the parliament. I suspect that this is not a provision to be found anywhere in the constitution and laws of Romania, now or then. For the second, assuming for argument's sake that the marriage was valid in Russia, if Romanian law does not apply in Russia than neither does Russian in Romania. In Romanian law the first marriage of Crown Prince Carol was invalid and the second marriage good. And what other law should govern the succession of the Romanian crown?
I do suspect that there is a great deal of republican spite involved in the decisions of foreign courts on the question, and political pressure from the current ruling powers of Romania in those of local courts. If it be argued that there was political pressure involved in the original decision too, what of it? One politically-motivated court overruling another doesn't seem like the kind of thing that should overturn the whole basis of Romanian succession. And, to reiterate, either only the younger son of the later Carol II was legitimate and therefore the elder barred from succession, or both were legitimate and the elder still barred from succession on other grounds. Legitimate or not, it is not possible to hold that the elder had succession rights. And since under the laws of Romania King Michael was legitimate he did.