sorry but not familiar enough with that program but it and the rest of my adobe products which have gathered dust on my computer the last good part of a decade (except photoshop and illustrator) have moved up the totem pole for me to play around with soon.
but i did google edge animate transparent backgrounds and came up with a number of hits, including you tube videos. it seems like it ought to be a built in feature to create a transparent animation but at least i think they might be useful to look at in case you haven't.
i've had this issue in my book development with off the shelf gifs, but the solution is easier as there is usually a white, black and transparent versions of these. or i have found a graphic i would like to use. i just choose the vector version if one exists and eliminate the background. in case there is no vector and it only exists as a jpg, one technique i use is to place the graphic in a picture frame (inspector, grraphic, stroke, picture frame). when it works visually it looks fine in my opinion. don't have enough knowledge as to whether you could use a similar trick.
hope one of these ideas works for you.