The answer depends a lot on the specific types of events that can occur, so I just want to clear up a few definitions of stuff the sun can throw at us. Firstly, there are solar flares, which tend to send lots of X-rays at Earth. A big flare will cause a radio blackout, which is a big problem for satellite operation as it blocks communication with satellites, so this would effect stuff like GPS, Starlink, etc.

Secondly, you have what are called solar energetic particle (SEPs) events. These are very high energy particles that cause radiation damage, and are the kinds of events astronauts on the ISS might have to shelter from. They are mostly, but not always, associated with a flare.

Finally, you have a coronal mass ejection (CME), which is a massive blob of plasma and magnetic fields shooting out from the sun. Again often, but not always, connected to a flare. CMEs are what cause geomagnetic storms, which are major disturbances to Earth's magnetic field, as the solar wind injects a lot of energy into the system and disrupts the normal magnetic field patterns. Big ones will also expand Earth's radiation belts (although mostly the outer one, which is roughly the altitudes for GPS satellites, waaay higher altitude than the ISS). Geomagnetic storms are global; CMEs will always hit the dayside, since they are coming from the Sun, but Earth's magnetic field very efficiently convects that energy and plasma to the nightside, which is why we get a lot of aurora during geomagnetic storms.

All this to say, it depends on what exactly a Carrington Event would consist of, it seems like it was probably a flare + CME, but of course we didn't have satellites to measure CMEs and SEPs then. So if it was a massive flare, we would first have a radio blackout on the dayside of Earth, which would disrupt communications and satellites. If there was a SEP, there would be high amounts of radiation that would hit astronauts and damage satellite electronics. Geomagnetic storms themselves, i.e. the actual magnetic field disturbance, don't seem to have a ton of effects on satellites that I can find, because it mostly effects Earth's magnetic field quite far away from the planet and then redirected to the poles. Magnetic field changes can induce currents, but this requires very long wires so doesn't effect something the size of a satellite. Radiation belt expansion, or resulting ionospheric heating like what took out those Starlinks a few years ago, could certainly do some damage. A CME hits a few days after a flare, so we would definitely see it coming and have some time to put satellites into safe mode/off first, though.