The bug removal device doesn't appear to penetrate Neo's skin, so there's no reason to expect a wound from the "claws":
Some redness on the skin might be expected, but not much else.
The "violent removal" of the bug through his navel is more difficult to explain, but it turns out that it's possible in the real world (our real world, not the one in The Matrix) to do surgery through a person's navel -- this is called a laparoscopy. According to the U.S. National Library of Medicine, the surgery is a fairly minor outpatient procedure:
Afterward, the incisions may be sore. Your doctor may prescribe a pain reliever.
You will recover for a few hours at the hospital before going home. You will probably not stay overnight after a laparoscopy.
Another source describes the procedure itself a bit:
The typical pelvic laparoscopy involves a small (1/2" to 3/4") incision in the belly button or lower abdomen.
...a laparoscope (a one-half inch fiber-optic rod with a light source and video camera) is inserted through the belly button. The video camera permits the surgeon to see inside the abdominal area on video monitors located in the operating room.
The size of the incision is on par with the bug that was removed from Neo, so the procedures are comparable. In real life the surgery is performed with general or local anesthesia, but it's possible (albeit very painful) to remove the bug without anesthesia.
You've noted that there's some blood and gore when the bug was removed, but in a real laparoscopy there's not much more blood than that. This video (WARNING: graphic) shows a laparoscopy done on a patient under local anesthesia and there doesn't appear to be much bleeding even when the laparoscope is removed:
The bug removal device probably vacuumed up most of the blood from Neo's navel and, since we don't know much about the device, it's possible that it can apply a clotting agent to the wound to further prevent bleeding (or, if nothing else, Trinity could have applied a clotting agent and/or bandage shortly after she finished removing the bug). We don't see Neo's abdomen after the bug is removed except immediately after (and even then it's not a great view), so we don't know much about how the wound was cared for. Neo might have also been given a painkiller off-camera.
Since it's possible in real life to perform a similar procedure without much blood and gore, and we don't get a good view of the wound and its care, it's within the realm of possibility that Neo's wound was too simply minor to notice. It's not even necessary for either the bug or the removal device to have special properties within the Matrix simulation to explain the minor wound.