Final answer:
After undergoing Billroth I gastric surgery, the client's symptoms of fatigue, numbness and tingling in the feet, and difficulties with balance suggest pernicious anemia.
Step-by-step explanation:
The nurse suspects that the postoperative complication the client is experiencing after undergoing Billroth I gastric surgery is Pernicious anemia. Pernicious anemia is a condition where the body is unable to absorb vitamin B12, leading to a deficiency of this vital nutrient. The symptoms of fatigue, numbness and tingling in the feet, and difficulties with balance are classic signs of vitamin B12 deficiency.