icot
Not in this case, i believe.
Well, an unmarried guy can light the Shabbos licht first because he is nohaig to not accept Shabbos until shkiyah.
But a woman should light the menorah first because she is nohaig to accept Shabbos with licht. She can of course make a tanayh to not accept Shabbos with licht but this isn’t recommended by M’B.
For a couple it wouldn’t be l’chatchila for the woman to bring in Shabbos with licht and then the husband to light Chanukkah (since he is being motze her), though it certainly can be done like that if necessary.
At least that’s how I learned it.
What do you say?