You’re absolutely right about how quickly some bacteria can replicate, but that depends on the proper substrate, ambient conditions, availability of nutrients, and any competition from contaminants.
What something like E. Coli can do in a well bioreactor is the ideal case, and even then most of what they produce is the bacteria themselves. On Earth this isn’t a problem at all, but as a means of husbanding every joule because you don’t know when or where the next one is coming from, I think it might matter.
It’s also probably a genuinely hard problem keeping your organisms viable without a constant supply of food, a means to get rid of mutants, or some hitherto unknown means of preservation that could handle the extreme time spans involved between “awakenings”.
Mainly my point there is that it doesn't seem reasonable to anchor advanced nanotechnology on the doubling times we're used to for industry. I don't want to guess just what to expect for early construction from a starseed arriving at e.g. an outer-solar-system carbon-rich moon -- but nothing like a human generation.