Welcome to the best forum around! The short answer to your initial question is yes, you can replace with just about any size engine you want. The long answer is.... it's a bit more difficult than you would think.
First off, you need to be looking at Kohler engines that were built specifically for Cub Cadet tractors. This is because the Cub Cadet Kohlers have specific crank shafts and flywheels that will take the Cub Cadet PTO, and allow you to bolt the drive system to the flywheel. No other Kohler engines have these adaptations.
Honestly, I'd be rebuilding the engine you have rather than repowering with another, as long as it can be rebuilt. The 12-hp K-series engine is easy to rebuild, and there are tons of blocks around if your block is damaged. However, you'll need all of your parts as mentioned above.
Most of the Cub Cadet tractors of that era ran the twin cylinder, 17-hp, KT-17 Series I engine, which was a piece of junk because it utilized splash lubrication, and they are well known for starving oil to one cylinder and blowing one side of the engine, often referred to as the KT-8.5. The worst part about this is that the engine would still run, and you could still drive the tractor around relatively well. The single blown cylinder is really not detectable until you load down the tractor, or try to engage the mower deck, which just kills the engine. The only reason this is bad is that LOTS of folks will sell you a "running" KT-17 engine that's really only running on one cylinder. Then, you'll find out that it's about $750 to rebuild that engine, and takes someone with a lot of Kohler experience because it's not like the old K-series engine where you can drop the oil pan and pull the head and do the rebuild. The KT series engines don't have an oil pan, and you have to literally split the whole crank case to do the rebuild. Easy enough, however the reassembly is VERY tricky and it's real easy to screw it up!
Another option that a lot of folks do, if you really want to build a beautiful machine, is go to Small Engine Warehouse and look up their Honda replacements for Cub Cadets. This will set you back about $1500, but their kit has all the adaptor plates already made up so you can bolt right in a 20 or 24-hp Honda engine. I know several folks who've done this and it REALLY makes a nice tractor.
Cub Cadets 682, 1811, 1864, Simplicity Legacy XL 4x4 Diesel with FEL, 60" mower, 50" Tiller