With a population growth rate of 1 % per year the population doubles every 70 years. 2685 years are about 39 doubling periods which means that the population will increase by a factor of 549,755,813,888 over this time and if we needed on Earth at the beginning we then need that many Earth-like planets or the entire Milky Way under my assumptions of 400 billion stars and one inhabitable Earth-like planet per star. This assumes no further or at least only negligible technology advances over those 2685 years, i.e. we do on the new planets what we did on Earth when the carrying capacity of Earth was reached, but which we assumed was the best thing possible at that point in time.
Exponential growth is really that bad and 1 % per year is a pretty high growth rate. We could never have sustained such a growth rate for extended periods. We only surpassed 1 % per year in the early 20th century, before that it was a lot lower. My entire point was to demonstrate that exponential growth with 1 % per year is so huge that it would grow beyond any realistic resource limits in pretty short time. Decreasing the growth rate by a factor of 10 to 0.1 % per year will only increase the doubling time by about a factor of 10 and we would still reach the carrying capacity of the Milky Way in relatively short 26,728 years. Stabilizing the population at some point is really the only option, exponential growth can not possibly work besides with growth rates that are essentially zero.
I'm obviously missing something in the figures you're using to calculate here, but how exactly would we consume all the galaxy's resources in < 3,000 years? We've been doing our thing on Earth for more than 3,000 years, and there are still resources available.
I'm not disputing that we consume far too many resources in a very irresponsible fashion. That just seems to me to be a very much higher rate of resource consumption than we've seen historically. Surely, if we were to develop the ability to colonize other planets, we wouldn't reproduce the same wasteful resource consumption patterns and history, because we'd have greater technology and, more importantly, improved philosophical outlooks on how we use those resources. Are you thinking we'd move 100% of the human population to each new planet and just set up the same wasteful shop we have till we exhausted the planet? I'd expect moving a smaller portion of the population to each new place, and setting up a far less wasteful colonization effort, that itself would continue to improve (hopefully).