He had an Hier and would have been a King in his own right if he passed on Morgase offer. Also it is well known that SHE neede HIM to solidify her claim to the Andoran throne ... Se was the weaker party for a decade, so, why if he was still the Hier to Laman, would he become a Trakand?

A man like Taringail would have moved back to Cairhien and set himself up a power base as the Hier as soon as Tigrain "widowed" him ... Unless there was some legal/published reason for him not to (ie. he had become a Mantear). Also, we have not once heard him refered to as Taringail Trakand ... Always Damodred, just like Galad. It seems far more likely that the opposite of what you suggest is what happened. Morgase needed Taringail to solidify the Throne and thus allowed him and his son to revert to Damodred from Matear upon their marriage.

I assume political machinations stripped them both of the Sun Throne, even if they had the best claim.