Huh - maybe I'm sleepy or something (it's a bit late) but, um,...
Are we making a mountain out of a mole hill, ie overcomplicating the technology required?
Let's take the concept of putting a satellite in orbit first.
We do it with a rocket launch and some carefully precalculated equations and a tiny bit of adjustment once in place.
You can you do that with a cannon. Just need to make it big enough.
So... extend that to a flight to the moon.
Same cannon, but fire a up pieces of a rocket, assemble in orbit and then, using radio to communicate burns (length, thrust, and vector) you can do all the computations before time or on the ground.
So all you really need is radio... or do you even need that if you increase the amount of math before hand? Fire pieces into orbits that will intersect, then build. Then aim for moon and fire thrusters?
Safe? Nah
Without any difficulties? Nah
Possible? Why not?
Seriously, if you did all the math before hand you should, in theory, be able to use that cannon to shoot an object into an orbit that decays into a slingshot orbit from earth into an orbit around the moon. Trickier than a 3-rail bounce and triple ball split in pool, but still calculatable. Now, getting back gets even more insanely tricky, but possible.
At least that way, you don't have to drag Babbages computer with you
Done this way, or with any other similar style of shortcuts, you could maybe get there in the early 1900's. At best. More likely the 40-50's as mentioned.
And that's with a lot of butterflies.
The only other way to meet the challenge would be to compress a few hundred years of scientific advances into the 66 year period. ASB's are the only way I can think of.
Hmmm one other possibility *might* exist. China. If it didn't undergo cultural convulsions in the 1800's but embraced steampunk Victorian Era British culture 100%... They might have the sheer population to push science much quicker *if* they could somehow share information as quickly and easily as we do on the internet...
Population because a certain % of your pop are geniuses. If an IQ genius of 180 arises in 1/1000000, if you have 6 billion, you have 60 of them. Get 'em together and science flies.
Literally.