I agree completely with Don, the 350 is easily capable of doing the work properly, it all comes down to the thinning of the paint. Your HVLP gun is meant to spray in a more industrial application, laying down much thicker volume and higher pressure. The paint passages are of much larger dimensions. A different airbrush may well not provide any appreciable change in performance.
Example: I've sprayed with large spray guns and later the HVLP models, have used enamels, lacquers and polyurethanes. Always shot at 30-40 psi, thinning was done to the requirements of "the eye." The paint manufacturers gave a recommended thinning ratio and psi, I made adjustments from there, depending on the object being painted, temperature, etc. When I ran that paint through the airbrush and it was properly thinned, I was able to spray at my customary pressure of 15- 20 psi, when doing hobby work.
Using an airbrush to spray that same paint on very small objects, required thinning MUCH more than when using the larger guns. I have used the Badger 350 and other airbrushes, I think for your intended use the 350 will do as well as any other. Try thinning to Don's recommendation of "like milk," after thinning place a drop on a vertical surface and see how it flows. If needed do the same with a drop of milk, then you'll see what consistency you're looking for. I'll bet you'll be surprised at how much you have to thin the paint for airbrushing.
Give a try at a really thinned batch before buying more equipment, a different brush will likely not do any better, but will lighten your wallet. Best of luck, I hope this will help you get it sorted.
Patrick