I'm doing some research about comet perturbations, and have noticed that the outbound aphelion is often different from the inbound aphelion, particularly due to the gravitational influence of the giant planets. However, as many of these comets pass very close to the Sun, I am wondering how much influence the Sun's atmosphere has on aerobraking the comet, further lowering its perihelion. What percentage does this make up, and have there been any comets which have been affected by atmospheric drag?
About 0%. If they experience drag by the Sun's atmosphere, they desintegrate due to tidal stress (internal strength of comets is ~Pa < kPa, e.g. Heinisch et al (2019)).
What does have an influence is the (increased) outgassing and mass-loss at perihel which is often not uniform - even well outside the Sun's atmosphere.
On the hand, gravitational influence of gas giants on comets is often not negligible. Their main gravitational influence is in the form of reducing the semi-major axis of the comet's orbit, thus turning long-period comets into short-period ones. Due to the comets low(er) orbital speeds at aphel and their long time spent at aphel, the momentary influence can be tiny but have major effect on the overall orbit of the comet. It's assumed that most if not all short-period comets were once long-period ones who at some point got their orbits altered by Jupiter or one of the other gas giants.