I don't think the telomere approach will work at all. If it did, we would have died out. Or rather, would never have existed. Short telomeres are a normal thing, happens to everybody (except the ones who die in an accident), yet nature "just makes it" so gametes are good to go anyway. Sperm from a 100 year old is, well, not precisely as good as sperm from a 17 year old in terms of numbers and mobility and such, but... whatever. It still works, and whatever comes out is a perfectly good individual with perfectly good telomers.
Now, polyploidy might be a better strategy. While it can be somewhat troublesome on higher animals, it's a well-established thing for plants, some frogs and amphibian stuff, a couple of marine vertebrae, and some worms.
If you have a population of, say, 2n males and 4n females, the offspring will be 3n. For a 3n individual, undergoing meiosis is, uh, troublesome. Because 3 doesn't divide by 2 so well.
So, that's that. No gametes, no offspring. Unless you do it like wild dandelion, which despite being unable to reproduce just tells you "Yeah, you know what, f... off!" and simply goes agamosperm, reproducing anyway. But for an animal, that's reasonably unlikely to happen.
The real challenge is that you want not 2 or 3, but 50 generations which are OK, so you would need to find a number which only results in an odd pairing after 50 generations. That's probably a tough one.
It's fiction though, so... got some leeway, I'd say. Might just say "poly" and not mention how many.