I agree the texture looks like a cast die. Maybe the clarity of the photos but of many things the stars on the obverse do not appear to be of correct. Denticles look weakly stuck with improper symmetry. First and foremost is what appears to be a bubble surface on the reverse next to and below the right wing. I would recommend you look at a PCGS, NGC or ANACS graded example of a 2 1/2 $Dollar Gold Liberty in UNC quality. Look at this sight for a start https://www.pcgs.com/Photograde/#/2.5Lib/Grades
I'd be surprised if it is fake. My dad collected a lot of coins and the vast majority he bought were already graded/sealed by NGC or PGCS. There are only about 10 in his collection (this being one of them) that were not graded and 8/10 were ancients.
Keep in mind, if real it's in nice shape but there are literally thousands of these in Mint State slabs. That - and the evidence you present of your father's efforts - makes me fairly confident of its' authenticity, but as a result even if real its' value isn't going to be sky-high. It'd need to reach MS66 to be worth as much as $1000.
Still leaning fake. Notice how in every one of these, the denticles that almost touch the one in the date kind of dip, so it is not a perfect circle of denticles. Yours does not appear to do that. http://www.pcgscoinfacts.com/CoinImages.aspx?s=7859
Also notice how some of your denticles are almost a trapezoid, where they are skinnier near the rim, and then get fatter. All genuine ones appear to be parallel.
I was looking through some coin auctions and this one seems to not have the "dip". What gives? Do you see it? Did I miss something? https://coins.ha.com/itm/liberty-qu.../1215-3236.s?ic4=GalleryView-Thumbnail-071515
Hmm, not sure. But the denticles still look different. It looks like a fake to me, and I buy and sell tons of gold coins. If I saw this one for sale, I would not offer to buy it except for melt value, if it was real gold. Toughcoins also buys and sells tons of gold, and also thinks its fake. Take our advice for what it's worth. If you get it graded, let us know the results.
Not all of them have that dip in the denticles, I noticed the dip while looking through coins last night but then found a few that didn't have it. On the ones that don't have it though it appears the first digit was always touching the denticles like in the one your link shows. This is perhaps why they abraded the bottom of the digit/denticle trying to get separation and that's why the dip appeared.
The wing feathers seem to be lacking detail. Very weird. It looks like it would have to be polished to lose that kind of detail if it is real. I hope it is. Good luck!