Hooookay,
But methinks that from this:
…others shall smell blood in the water anyway… 
If a critter like Nessie reproduced asexually, then significantly fewer (not just one, but fewer) individuals would be required to maintain a population, but I’ve never seen numbers on minimum viable populations for something that reproduces asexually only.
Asexual reproduction doesn’t solve all the problems involved, though. Completely unaffected by mode of reproduction are catastrophes, disease, loss of genetic variability, predation, competition, and food supply, all of which could drive a small population to extinction.
Potential threats to the population like demographic fluctuation, inbreeding, and genetic drift are reduced, but not eliminated, if the critter reproduces asexually. Only demographic fluctuation is effectively put to rest as a threat to the population (even if you get them down to one, the birth rate can still rise). Asexual populations require genetic variability too, especially in order to respond to long-term environmental changes, like deglaciation. IIRC, the overwhelming majority of asexual organisms can also reproduce sexually in order to reap the benefits gained from mixing up the gene combinations frequently.
The ability to reproduce asexually does not solve the challenges that face very small populations, although it may allow them to last longer than would otherwise be possible, I can’t really be sure.
Changing gears…
It appears that my thoughts on Nessies living in the great deeps would not be possible, as Daniel has posted some quotes that make the link between the ocean and the Loch unavailable for transit back and forth. If there is no useful link to the ocean, then I don’t see how a population of Nessies is even worth consideration.
-Steve