When a question is scrutinized too intently the obvious can become elusive. The particular attributes of the individual animals in an isolated population that successfully reproduce determines which attributes are available in subsequent generations.
Mere millenia aren't enough to expect mutations to give rise to a significant number of new attributes. That implies that the bears on your island will be gradually reducing the diversity of their genome. Individuals express a subset of their genes in response to their environment. Unused potential isn't lost instantly, or even rapidly, except when its presence reduces reproductive success. Expect the bears characteristics to remain within the available diversity they arrive with.
The answer to the question "do I get giant cave bears or pygmies after 1000 generations?" depends on what effect the size of an individual has on its reproductive success. Its more common for the pressures on an island to favor smaller individuals, but giants like the Irish Elk and the Kodiak Bear result from environmental pressure favoring larger individuals.
In the hypothetical case, the environment is a benign climate with a predictably limited diversity of food sources and habitats. A large highly adaptable omnivore like a bear is likely to be so successful competing for food and habitat that the ecosystem collapses and the bears die off in a relatively brief time scale. An island ecosystem is fragile.
The habitats and species these bears share the island with would be more likely to remain in balance if some sort of an unusual circumstance forced the bears to compete with each other for an exogynous food source rather than exclusively with the other inhabitants for indigenous resources. For example, Kodiak bears compete with each other for fishing spots for the salmon run that is a huge fraction of their annual caloric intake. Its safe to assume that anything indigenous that a bear that didn't fish successfully could eat instead is likely to be eaten.
The argument that a certain population size is needed to ensure adequate genetic diversity is unavailing. Its an anthropic argument, i.e. that some population must be adequate for a species to survive/fail because the species isn't/is extinct. It only works a-posteriori, i.e. after the fact. There were an enormous number of passenger pigeons just a few decades before the last one died in a cage in a zoo. Tens of millions of individuals. That there wasn't anything in their genetic pool that would let them survive the loss of the continuous swath of old growth forests was self-evident after it happened, not before. Environmental pressures determine which individuals survive to be able to reproduce, and mating preferences tend to favor healthy individuals. If being smaller makes an individual more likely to survive and be healthy, larger is less likely to be available in later generations and vice-versa. How many individuals are needed to ensure sufficient genetic diversity for a species to survive some hypothetical environmental upset in the future has no role in the process. The genetic availability of un-expressed traits has no role in the process.