Well, the subjects you are wanting to view mostly require aperture to show off. So going down to a tiny aperture will reduce your ability to even see a galaxy or nebula, clusters are different and will show up, but what magnitude stars that will show up will change which will alter just how deep you can go on a them. Also, the subjects you are referring to are mostly low contrast brightness, so again, aperture and a dark sky is what it takes to see them ideally. Going to a tiny aperture refractor will not enhance this, but rather, you will lose subjects. But that said if you're not using your current system, you will not see anything, so whatever gets you out there will be better if it means you spend more time out there under the stars. Because most of the subjects are not bright point light sources, they will not suffer from CA degradation, so you don't have to have purely apochromatic optics for ideal viewing. Some clusters will have CA on the brightest stars at medium to high magnification, but not at low magnification as much so less noticeable.
So really, a 150mm F5 achromatic doublet would be a great refractor for your goals coupled with a dark sky. Or at the least, a 120mm F5 achromatic doublet (these are cheap and great for this).
Otherwise, frankly, your 200mm aperture is more ideal for what you want to see, than a small aperture refractor (say 80mm to 120mm to keep the size/weight reasonable).
$3k is a very healthy budget for visual. Your subjects you want to see are best seen with a dark sky as they are heavily effected by light pollution due to surface brightness (galaxies & nebula); clusters not so much, but still a bit depending on how deep you want to go with them (which takes aperture to increase minimum magnitude of stars you can pull in). So getting a high quality 102mm APO for example will not do a better job at all on these subjects, magic glass won't help, apochromatic glass will not help, a dark sky and large aperture is what will bring out these faint objects you're saying you want to see.
So if a 8" dob is too much, here's what I would propose:
Instead of getting a small scope, maybe consider what would it take to allow your bigger aperture scope to permanently be setup so that you can use it at will without setup time and without having to move it. Is this possible? Think observatory/shed/TG365 cover, etc. What would be possible? If you could do this, you could use a big aperture and enjoy those faint fuzzies without the physical issues of big aperture scopes. I would start thinking this direction, if you can with property, etc, obviously this may not work depending on the land/property issue or where you live, etc. But if you have the option to have something permanently setup, I would pursue this rather than trying to get by on faint fuzzies with a tiny scope, unless you have supremely dark skies.
You can find and see the faint fuzzies with a small aperture with experience and a dark sky. But you still won't see their entirety, you'll just see the bright cores or brightest section, which is entirely different than actually seeing the full structure. An example would be seeing the faint core glow of a galaxy in a small aperture and just having to be ok with that you found a galaxy but can't really see much other than a faint glowing core, versus using a larger aperture and seeing that same core but also being able to see spiral arms around it and actually see the structure. Some are ok with the "idea" of seeing an object, and checking it off a list, others want to actually see stuff and not just imagine what it looks like other than the fuzzy glow and knowing what it is and it being mostly a thought experiment. You gotta sort out what you want out of this and what your expectations are.