You are doing the right thing by trying to supplement her with puppy formula and alone time with Mom! 7 puppies is a BIG litter for a Yorkie!! You would be wise to have your Yorkie on Calcium supplements like a Tums w/ Calcium 2-3 times a day during lactation/nursing. Small breeds are very prone to a condition called Milk Tetany or Hypocalcemia due to having to produce large amounts of milk for their pups!
For your runt, the best option is likely going to be that you continue to supplemental feed the pup several times each day, but allow it to have access to the mom in private several times a day. Not only for an opportunity to nurse naturally, but to get "mommy time" During the 1st week you should try to feed the runt every 2-3 hrs during day up until midnight, and then at 4:00 and 8:00 AM. If all is going well, reduce feeding to every 4h during the 2nd week, and to every 6h during the 3rd-4th weeks. The exact schedule will vary with the amount the animal takes in at each feeding and its state of health. It is also important not to disturb the animal if it is sleeping. If the animal has a strong suckling reflex, a small animal nursing bottle or doll's baby bottle may be used. Pierce the nipple with a hot needle so that a drop forms over 1-2 seconds when the full bottle is tipped upside-down. A syringe or eyedropper is not recommended, but can be used in an urgent situation where a bottle is not available, but it is essential to deliver the liquid into the mouth slowly to prevent aspiration.
Formula (milk replacer) product recommendations should be followed with respect to quantity of supplement to feed. You can tell that the stomach is full from feeding when the belly is distended or the animal turns its head away from the nursing bottle and squirms. Constipation or diarrhea may occur with formula feeding. If constipation occurs, use a gentle warm water enema. If diarrhea occurs either 1) reduce volume to previous level if volume appears to be too much; or 2) keep volume the same (to ensure adequate water intake) and dilute concentration with water by 1/3-1/2. Semi-solids can be added to formula in week 3, and the animals can be introduced to lapping in weeks 3-4. Animals can usually be weaned off support by weeks 5-7.
IF the mom has not been lavishing alot of energy on this pup, she may sense that it is a runt and is trying to distance herself from it. Often times the runts do not get stimulated to urinate/defecate and will begin to get bloated and distended causing them to sleep excessively, have poor appetites, and cry. If you are not already stimulating the runt, you can take a warm wash cloth and gently massage the perineal and vulvar areas with your covered fingertip. Within seconds, the pup should urinate and defecate. This needs to be done prior to every feeding for a minimum of two weeks, sometimes longer. If she has a poor suckling response, you may have to take her to your vet where they can show you how to tube feed her on your own until she gains some strength back and can nurse.
You should be weighing her on a gram scale/food scale daily. With each day, the pup should be going up in weight...not going down or staying the same.
The pup should be kept warm at all times. If the pup gets chilled, it can become sluggish and not want to nurse.
It is sometimes helpful to rub a finger tip of Karo Syrup across the tongue of a pup prior to nursing to get a sugar load in their system in prep for nursing/feeding.
At some point, you should likely have the runt examined by your vet to determine if there is any evidence of a congenital heart defect or other possible anomaly that would explain her condition.
Once again,considering the size of the litter, I would just say that 7 was NOT her lucky number!!
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Best wishes to your and your pet, Dr. Smith