If a women is due to go to the mikva but she is at her inlaws house in another town and there will be a friday night meal with an entire family, is she obligated to go to the mikva anyway even though it will be an embarrassing situation? Does it help that the couple has a boy and a girl already? Also is there an issur to delay going to the mikva for any embarrassing case if both the husband and wife agree to push off the mikva to a better day?
As mentioned in the previous answer (https://dinonline.org/2011/08/15/posponing-mikva-because-of-tzenius/), it is better not to postpone.
Although we do not rule that there is tevilah bizmanah today, it remain lechatchilah to go at the first possible opportunity. It is true that the reason for this is given as for the purpose of piryah u’reviah (Yoreh De’ah 197), but I have not seen poskim make the distinction between those who already have children and those who don’t. Another reason poskim mention is for the fulfillment of mitzvah onah.
However, under uncomfortable conditions, when both agree, and when delaying will be beneficial for the relationship itself, one can (and even should) be lenient and delay the tevilah.