The direct answer to your question would certainly be more than fairly difficult. I'd go with something between extremely and exceedingly difficult, but not impossible. It will require an incredible amount of luck as well. The fact that your grandparents on that line were second cousins once removed is going to make this considerably more difficult, especially given that their common ancestors were Minnie's in-laws.
I see you've already got a reasonable start on your DNA confirmations, but you're going to need to take that to a whole different level. If you haven't already done so you'll want to have your father's DNA at all of the major sites (you really need to work with your father's DNA, if at all possible, as it gets you at least one generation closer): Ancestry, MyHeritage, FTDNA, 23andMe and GEDmatch. This will require triangulation so Ancestry won't be of much help at that distance, though it can provide some good clues.
Next you'll need to find some matches with descendants of Minnie. This will mean building Minnie's tree out and down as thoroughly as possible. Depending on how intent you are, you might consider contacting some of her descendants and attempting to get them DNA tested. Then you need to start looking for triangulated matches with any of those descendants, particularly those triangulated matches that are smaller than your match with Minnie's known descendants as those matches are likely (but far from certainly, at this distance) to have Minnie's ancestors as common ancestors with you. What you need to do is find two triangulated matches that share a common ancestor who meets the requirements to be Minnie's father. This is a real stretch, but who knows, you might get lucky.
I've been searching for my mother's 3rd GGPs this way for several years now and am only now starting to get some reasonable ideas of where they were from and who they might have descended from. In the process I've created a spreadsheet that I update about every six months that includes the segment data for all my mother's DNA matches from all the sites that provide that detail. I've also created many thousands of profiles here on Wikitree trying to build out the trees of those DNA matches who might triangulate with matches known to descend from her brick wall 2nd GGPs.
In the end I'd recommend thinking of it as a journey on which you'll undoubtedly learn lots about your ancestors and you'll contribute to building out the tree. And if you're really lucky you might just find your 4th GGF some years from now.
Good luck and have fun!
