You've never had a gate-area TSA check at the transit point that happens to be the US gateway for your international flight? I've seen this happen (even when CBP is not involved doing a gate-area/jetbridge checking of passengers) and they check for ID sometimes.

Also, there are some routes, such as California-ORD-CPH, that involve clearing the TSA TDC at the transit airport screening checkpoint for the same-day connecting flight to CPH. The TSA at ORD, for example, generally checks for ID on this routing to CPH. Much the same goes for many Star Alliance trips that have California-EWR-CPH routings.

If the OP arrives at the airport early enough and takes a more direct routing like SFO-CPH by using the LPR card with the TSA and by using this emergency travel doc with the airline and CBP/passport control, then way more times than not it will work just fine and without issue. It does help if the names on all the docs line up and the Swedish consular officials are available to be contacted if there is doubt about the legitimacy of the document. My bet is that once the airline notice she's Swedish (and by the way she speaks in Swedish) the airline won't cause that much issue -- at least if SK's own employee/management is on-site.