Well, there is a sorta information-theory theorem regarding this (called the Gerzon-Craven limit) that was published long ago by Michael Gerzon and Peter Craven back in the late 80s. i can't get a direct link to the paper reference, but type in "Optimal Noise Shaping and Dither of Digital Signals" into the AES Convention paper site. this paper is a quarter-century old. i just discovered that i have a camera copy of that old AES preprint and can send it to you if you send me an email to email@example.com .
I also have a copy of a paper making reference to the Gerzon-Craven limit On psychoacoustic noise shaping for audio re-quantization .
There is an interesting title regarding this called Approaching the Gerzon-Craven Noise Shaping Limit Using Semi-Infinite Programming Techniques. i don't have a copy of that and do not know how to get a free copy.
Essentially, because of the Shannon Channel Capacity theorem, the answer to your question is "yes almost, theoretically", as long as you don't care about the magnitude of the quantization noise outside of your original 22.05 kHz baseband. this is because the bit rate at 16-bit, 44.1 kHz is about the same (maybe just a teeny bit more) as the bit rate at 3-bit, 192 kHz and you can make an information-theory case that, because the information rate of the two cases are about the same, then the S/N ratio in the band of interest is also about the same.
But the real answer is "no, practically", because of the difficulty in constructing a 3-bit noise-shaped modulator with such extreme specs and that remains stable. with a higher sampling rate than 192 kHz, the specs get less extreme.
If you have a working system (what's it coded in, MATLAB?), i would be interested in your noise-shaped modulator. what order it is and how well it behaves. sometimes these nasty high-order ΣΔ modulators behave badly in the presence of silence and we get, what are called, "idle tones". if you think you have a working solution, i would be interested in hearing how it works for very low-level input.