As an EB-5 investor, how can I leave the U.S. without receiving my AP combo card and not jeopardize my EB-5 application?

What did your attorney state, you should follow that advice specifically and all attorneys would state that one should not travel without the card. If you take the risk to travel and await the mailing of the card, that is a risk to have the application denied.

You do not want to leave the U.S. without the travel permit or you will be abandoning the I-485 application (green card). If urgent need to travel, you can call the USCIS Contact Center and arrange an INFOPASS appointment for an emergency travel permit (That will cancel the one that is pending) with proof the emergency.

If the AP card has been approved, technically, you can leave but its dangerous to do so without the card. If it gets lost in the mail, he will not be able to enter unless he has a valid H-1B (or L-1 dual intent visa). Do not risk it.

Your husband needs the AP card to reenter. Perhaps declare the card lost and request a reissuance?

You must have your AP card in your hand before you leave the U.S., or you will be abandoning your I-485 adjustment of status application. You cannot depart and have someone send you the card abroad. If it is an emergency travel, you can make an appointment with the local office or call USCIS and ask for a discretionary expedite of the AP document, after providing proof of the emergency (these, however, are rarely granted, unless in a true emergency).

An adjustment of status applicant who departs the United States without an approved advance parole document automatically abandons the adjustment of status application unless that person has an H or L visa.

If he leaves without advance parole, he abandons his application for adjustment of status. He will need a visa to return to the U.S.