I completely understand you; however, you have to understand that as an asylum applicant (with no asylum grant) there is no guarantee they will let you in... they may or may not...
If you have a good reason, in which you have, they may let you in with no problems... but like I say I never heard of a similar case that they let the applicant in...
This is what CIS says about it:
"Asylum Applicants: An asylum applicant who leaves the United States without first obtaining advance parole shall be presumed to have abandoned his or her asylum application. Advance parole (see: USCIS Form I-131) allows certain aliens to return to the United States without a visa after traveling abroad. Asylum applicants must receive advance parole before leaving the United States. Advance parole does not guarantee that the alien will be paroled into the United States. Rather, the asylum applicant must still undergo inspection by an immigration inspector from United States Customs and Border Protection (CBP)." quoted from USCIS website
You have 50/50 chances... make an infopass appointment and ask those questions and hear what they say about it...
As far as the UN refugee status... I am not sure how easy or how fast that is... but I heard that its very complicated and takes a long time.. anyhow...safety of family is always worth it...
If I would be you, and the situation with my family is really really bad, I will try to get an advance parolee -if they approve it-, then, go to the third country, save my family, do the UN refugee status for them, and come back to the US...if at the end they dont let you in into the country... you can go back to the third country that you were and try to get the UN refugee status for yourself too... it may not be the same as having an asylum grant in the US, but at least you have your family safe.... that's just my opinion...
But definately make that infopass appointment....
good luck...