Enoki
SOC-14 1K
Not as doable as you'd think.
First, your crew estimates are far too low. Three hundred is just for cruisers, the capital ships quoted require much more. Much more like 4K for a Tigress and 1K to 1.5K for the others. And we haven't counted frozen watches yet.
Service personnel, the 'teeth to tail" ratio, are going to be perhaps ten times the number of crew.
Then there's training, another factor that's hard to stuff into a spreadsheet macro. Yeah, during WW2 the USN was able to take hillbillies and turn them into radiomen and radar operators, but what was the TL "gap" in question? One or two? Are you going to take a kid from a TL 4, 5, or 6 world and train him in a reasonable amount of time to operate, maintain, and repair TL 14 and 15 equipment?
Then there's distance. Are you going to recruit a kid from Lishun to serve aboard a BatRon in Daibei? You're going to man your sector fleet from the sector in question.
You're going to need many more men than you think, you're only going to be able to recruit them from worlds above a certain TL, and you're only going to be able to ship them so far.
Go ahead. Build those 1100 ships in one year. Now man them.
Okay, let's say it's 10K per ship average. Most of those ships would be smaller ones, not massive battleships or such after all. That'd be 11 million personnel total out of a population of say, 10 billion. That doesn't even reach 1% of the total population.
If we say the ships require 10% of that total, that's 1.1 million crew. Again doable on a population of billions.
As for TL... It depends on how recruiting is done. It could be entirely possible that lower TL planets are targeted for recruits at about age 10 to 12 who are then placed in military schools off world until trained at 18 and they enter the service. Low TL worlds with large populations might be happy to have this happen as they eventually get skilled high TL people back potentially. For those chosen, and their families, this might be a way to rise socially and pay the bills. Many Second World militaries are seen that way today on Earth. Even top tier militaries are often seen as a way to get a trade or skill and get ahead for people at the lower end of the social spectrum.
You assume that the Imperium has a system and political problem with recruiting similar to Western nations on Earth today. It is entirely possible they use a different system. In ancient times, and even well into the industrial age, many nations coupled long term military service to other desirable outcomes for those who served.
You put in 20 or 30 years starting out at a military school at age 12 and you get a small plot of land back home, or a big bonus for retirement, or a decent stipend. Your pay might be several times what you could earn back home.
Now you have professional long service personnel that literally grew up learning how to operate ships rather than draftees. Given you have a population base that will easily support millions of people in the service and amounts to maybe 1% or less of the population, the service can be selective about recruits, keeps them in service for years, and has the time to train them thoroughly.
So, you take a smart kid of 10 who can read, write, and speak in a language the Imperial Navy will accept from a TL 5 world and begin training him at a military school to be an operator / technician in the Imperial Navy. By 18 when he is put into the service, he's a skilled and competent crewman.
The same goes for officers. Bring back midshipmen in a big way. Start with kids from more affluent families and let them grow up in the Imperial Navy.
This system makes sense given the relative slowness of communications and travel between systems. Such a system takes a long view of military service rather than it being a temporary thing.