yytz
“when a goy wants to convert but is already married to a Jew. We don’t urge him to immediately leave the Jew (although that would put an end to some aveiros),”
I dod not believe that is correct. when the goy converts he has to accept all mitzvos. He cannot remain with an irreligious woman who will not keep halcha. Although your reasoning may be a good one. A Ger has to be mekabel the entire Torah.
On the other hand, as I quoted from Rabbi Resiman besehm R’ Moshe “Nur ah Ger niz mekabel Zayn kol hatorah bevas Achas” A Baa lTeshuva should keep as many mitzvos as he can if leaving his goyish Wife isnt possible of course he should still keep kosher, wear tefilin etc.
and As ZD said these conversions are generally done, as Halcha says they are good once done.