Terminate solid twisted pair to UBNT in-wall AP?

I'm thinking about installing some UBNT in-wall APs into an existing environment with low-voltage boxes, wall plates and solid-core twisted pair cabling.

What's the best way to get the solid cable connected to the 8P8C socket on the back of the AP?

Options that have occurred to me so far:

• Snap the existing keystone jack free from its wall plate, stuff it into the box with a 6" patch lead.
• Re-terminate the wire into something like this toolless plug
• Re-terminate the wire into something like this (2 pair only - I'd probably shop for a 4 pair option) pigtail connector

What's the right way to deal with this?

Do I need to shop carefully for a punchdown solution that matches the gauge of wires in the wall, or is 110 punch stuff a one-size-fits-all soft of deal?

The plan includes pushing PoE+ down this wire for passthrough to hardwired PoE devices downstream of the AP.