May not be much help, but I use a simple zigbee button as a doorbell, and have a smart lighting automation tied to it. I believe smart lighting automations run locally as long as all the devices involved are using stock DTH’s, so the lighting response is near instant. The automation I use for the sound portion of the doorbell uses Echo Speaks, which is cloud to cloud, so it sometimes has a bit of a delay, and doesn’t work at all if the internet or cloud is down. I keep the lighting and sound automations separate so that the lighting is always instant, even if the internet or ST cloud is down.
Doesn’t include video or utilize your existing SkyBell though.
As @JDRoberts mentioned above, having something else like a motion sensor triggering a smart lighting routine running locally would be another option.
Depending on how much tinkering you like to do and how “hacky” you’re willing to get, you could potentially take the SkyBell apart and tie a dry contact sensor (or hacked open/close sensor) to the button in the SkyBell, and then use that to trigger a smart lighting routine.