Thanks for the additional information. It is helpful.
There are a couple of things that could be an issue. With bulldogs you can have some abnormalities of the palate and other airway issues found in the breed. This may be a problem with the pup which is leading to him not being able to nurse well. Read more on this here:
Another possible cause is herpes. With herpes a pup will often cry excessively, have bruising or stomach discomfort, and or greenish yellow fecal matter. They do tend to have a loss of appetite as well. This strikes pups too young to regulated their own body temperature. Prognosis is not good for pups with herpes when pups are under 3 weeks of age. Some breeders will try to keep the whelping area very hot to help combat the herpes. You can read about herpes here:
A heavy worm load can lead to a pup who fails to thrive as well. Mom and pups should be dewormed every 2 weeks starting at 2 weeks of age. Don't deworm the pup until your vet has given him a quick check over. Now with bottle feeding you have a much higher chance of the pup developing pneumonia from aspirated milk so be pretty careful when feeding. What I usually do is remove the other pups and let the non thriving pup have first choice of nipples for nursing and only let the other pups nurse once that pup has drank its full. Often putting a dab of karo / pancake syrup on the nipple will stimulate the pup into sucking and boost their blood sugar levels as well.
This site goes over ways of helping save fading puppies.
I would have this pup seen by your vet as soon as possible.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer positively so I am compensated for my time.