Peter, first up, best of luck. If you have patience and money, you will have options. But the requirement is atypical for genealogical purposes, and the options may not be readily apparent. You're correct on two points, though: AncestryDNA can't help in this case, and the yDNA from your dad won't shed much genealogical light that your test can't.
The one thing I would add to what Robynne and Paul said is that one genealogical testing company, Family Tree DNA, possibly shouldn't be ruled out immediately. The company that owns FTDNA is Gene by Gene, Ltd. (the holding company formed in 2012): same administrative offices in Houston, and they use the same lab, the Genomics Research Center, built by FTDNA circa 2007. By around 2009, all of FTDNA's testing had moved to this lab; it's the one where tours are provided in conjunction with the annual FTDNA conference.
Ancestry/genealogical testing is done under the auspices of FTDNA, but the lab, through Gene by Gene, also provides testing for clinical and research purposes, including whole exome sequencing (which you don't want) and whole genome sequencing (which you might want). The latter is performed by the lab only from a 3cc-5cc blood sample.
The full-genome sequencing ain't cheap: about $3,000...which is still less than a third the cost from just a few years ago. It can be had for less via other sources, but none (to my knowledge) are done at a lab and by a company that is also a major player in genealogy testing. The testing resolution averages 30x, which about standard, and the results are provided as a raw data file (the type known as "fastq," specifically).
For genealogy, we may never get to the point of standardizing on full-genome sequencing because the data files are huge and therefore are not easy to compare for matching purposes without a big-time upgrade in currently available IT architecture and infrastructure, and because a vast majority of the genome isn't very useful for genealogy: all humans' DNA is over 99% identical. But everything we currently test for genealogy--plus what we might add to test in the future--will be in that full-genome sequence.
The current genealogy tests look at certain sets of loci along the genomic map where SNPs (single nucleotide polymorphisms) of interest are known to occur...these are pinpoints of variability that include locations on the chromosomes where geneticists think much of that less-than-1% difference among humans can be found. The loci are uniquely keyed to the map of the human genome by what are called reference clusters, or "reference sequence IDs." If you open up the raw text file of a current genealogy DNA test, the column you see with all the digits prefaced by the letters "rs" are these map identifiers of the particular locus tested. For example, rs4477212 may be one of the very first ones shown on that list of over 600,000 lines (depending on the testing chip used) because it's near the very start of Chromosome 1 at base pair number 82,154 (in build 37.1).
Point being that if you have a couple of billion of a person's DNA base pair "letters" identified, you can create the subset of about 600,000 letters that could then be formed into a valid "kit" that can be uploaded to GEDmatch, FTDNA, MyHeritage, or elsewhere. This is somewhat analogous to what we've been seeing in the news regarding some of the recent cold-case solutions using "forensic genealogy": the unknown perp never took an AncestryDNA test looking for cousins, but forensic DNA data taken at the crime scene can be translated and formed into a "pseudo-kit" for uploading to GEDmatch, for example.
Since the same lab and same company does both types of testing, it might be worth looking into full-genome sequencing from Gene by Gene: https://www.genebygene.com/pages/research. At the very least, it may be worth a phone call to explain what you need, and to see if they have a mechanism to translate that whole genome data into a subset that can be the equivalent of an FTDNA Family Finder test. I've never personally dealt with the specific situation, but something tells me it wouldn't be the first time they or Dr. Rachel Beddard, director of the Genomics Research Center, have encountered similar inquiries.