I so wish I knew more about puppy care. I hope Ghilly comes online soon, she may be able to advise you. Or you could try asking your question in the "Ask a Vet forum" (there are many vets there who could help you) There is a small charge for advice on that forum, but it may be well worth it.
It sounds like you are doing your best with the little one. She has to be the runt of the litter, and you are giving her a better advantage, trying your best with her. I guess it's Nature's way for the mother to kind of ignore the sickly frail ones, as the stronger healthier ones push to the front more. In the wild, the runts would very often die.
However, I feel that if you could see your way to taking this little pup to the vet, it might be advisable, just in case it has aspiration pneumonia. If so, it may not live very long if it doesn't get veterinary support.
This puppy needs a trip to the vet ASAP to see what the problem is. Sometimes it's just a case of a puppy being the runt of the litter and not having the size and strength to push past its larger, stronger littermates to fight for a nipple. But sometimes there is a medical cause for a lack of growth and thriving, and this possibility needs to be addressed so that you know what you're dealing with.
If she appears to be having a hard time suckling, it could be that she has a cleft palate. This is not uncommon at all in puppies and it might be a case of having to find alternative ways to feed her until she is large enough to do surgery to correct it. Porto-systemic shunts (liver shunts) can also be responsible for a lack of growth in puppies. Depending on the severity of the shunt, this can also be accompanied by other symptoms like diarrhea, vomiting, and even seizures.
Briefly, when the puppies are in the uterus, the mother's liver takes care of filtering their waste. The fetal blood vessels allow blood to bypass the not-yet-fully-developed fetal livers, but as soon as the puppies are born, these vessels close off so that the puppy's liver can process its own blood. Sometimes the vessels fail to close, and the blood is misdirected, or shunted, AROUND the liver, instead of through it. When this is the case, the blood is not filtered as it should be, and the metabolic wastes are not adequately removed from the body. Each case is individual, so it's impossible to say, in general, what the puppy's chances are, if this is the case, without a complete medical evaluation. What IS definite, however, is that no matter how minor a shunt may be, it should always be considered to be life-threatening, and should be treated as such.
There is also something called "fading puppy syndrome", where puppies just don't thrive and eventually die. Nobody knows what causes it and nobody knows how to stop it. Three weeks is a bit far along for fading puppy syndrome, but it's possible that it could be this as well. The only way to know for sure is to take the puppy to a veterinarian for an examination. Otherwise, it's all just guesswork as to what the puppy's problem might be.
Please keep us posted and let us know what the veterinarian said. I'll keep good thoughts for your puppy and hope that it turns out to be something easily remedied, like the puppy just being too small to box his way in there among his bigger, stronger littermates. If that's the case, you can step in and hand-rear the puppy and all should turn out well. Again, let us know what the vet says. Good luck!