Another poster had the right idea when he simply said, "leverage". One tip, that I almost hesitate to share, is using two wrenches, end to end. Sometimes when you have space constraints it's your best option. You put the box end of the correct size on the bolt head. Then (and this is the tricky part) you take a second wrench and slip its box end over one jaw of the open end of the first wrench. It takes coordination. You need to put the second wrench over the jaw that is facing the direction that you are going to pull. The two wrenches will lock together as long as you maintain tension. It's very easy to hurt yourself doing this, so I'd strongly advise not hurting yourself. Otherwise, "knock yourself out".
Noted on the impact wrench — my plan is to try an air wrench. Hopefully my portable compressor (1.4 gal, 2.4 cfm @ 90 psi) can power it for at least short bursts.)
I have a small battery power impact wrench. It came as part of a combo cordless drill and impact wrench set, but its “impacts” are pretty weak — I only use it on small stuff I’m trying to cajole loose, such as a #8 screw at most. I remember another friend had a hefty battery power wrench with big honking ‘ol Ni-Cad batts. I’m pretty sure it would be too big, as you say.
My mechanic friend said they run into low profile bolt heads sometimes, but (being as they are a fully equipped 4-bay shop, much as you describe) they are only occasionally a problem. Why Ford would use such here is hard to say - as a tiny part of a weight reduction program is the only reason I can think of.