There are many high-throughput products that perform roughly this function of finding karyotypic variants. Bionano sells an optical mapping product that claims to do this, I have not directly inspected their viability; their tech is slightly complex.
Hi-C-seq is rather simpler and there are several competing firms that offer products that perform karyotyping:
I am sure that there are other methods including mate pairs and even plain whole-genome shotgun sequencing that could do ok at this task.
The Hi-C companies sell relatively inexpensive kits for library preparation, and some will also sell analysis services at fairly high throughput.
All of those methods should outperform mate-pair sequencing (which I think you call MP-seq) significantly, due to the relatively high rate of chimera formation (on the order of 0.1-1%) and generally the lower signal of mate-pairs, which are restricted by the insert sizes chosen. All of them should find the large karyotypic events found by FISH, ideally with more precision and sensitivity.
Traditional karyotyping services (such as FISH) run ~\$11000 per diagnosis, whereas the high-throughput methods I note (not the ones in that paper) probably are closer to ~\$500-1000/sample for preparation + ~\$500-1000 for sequencing plus maybe ~\$2000 / sample for analysis costs with a vendor (at a guess, for many samples). The companies in question may also do your sample preparation at a significant increase in cost (an extra $2000, perhaps). This would be a total of ~\$4000 / sample, as a high estimate.
Mate-pair costs would be cheaper at the sample level but more expensive at sequencing and analysis, not sure about the exact numbers. I don't think there are providers offering the package as a service, as it may not be economically viable, but I could be wrong about that.
I don't have refs for these prices, the providers themselves are the source of truth for that. Any of the companies above would happily prepare a quote for you.
If my very rough math above is correct, any of the sequencing methods should be cheaper, except possibly for Bionano (which is still probably more precise).
I'll note again my conflict of interest with Phase, though I don't work there anymore and I don't speak for them.