By what mechanism does the sun shift its orbit? How does it speed up and slow down to account for the dramatically different orbital sizes?
Unknown.
Need worries. The sun doesn't move like that model shows. It can't.
If the FE'ers could settle on a map - figuring out the motion of the sun would be fairly easy. We know that the sun crosses 15 degrees of latitude per hour - starting at 12:00 GMT at the prime meridian...then all you need to know is the longitude at which the sun is vertically overhead for every hour through the year and you'll be able to plot 24 points for every day and 6,360 points for the year.
Finding out the positions where the sun is vertically overhead would be a painful exercise - but you can just cheat and assume that the FE sun is somehow programmed to behave *precisely* as if the Earth was round...which is a surprising coincidence - but must be true or lots of people would have noticed by now!
I really can't be bothered to do it...but if you wanted to - you doubtless could.
However, we can easily to a few specific places without much effort: eg we know the sun is vertically overhead at the equator on the two equinox days - so we know it follows the line of the equator on that day. Plot those points on the bipolar map and start thinking about the places where the sun must be setting on the northern or southern horizon...and you'll soon understand that the bipolar map is junk.