Other than cutting that part of the branch off, is there any way to get rid of the borer before it infests my tree?

No way to get the borer, if it is because it is inside and protected; the sap is the tree's way of trying to eject it. A systemic insecticide, like Imidacloprid will kill it; but the damage will still be done and should be cut out and discarded or burned.

I forgot to mention that the sap is rock hard now, so would that still mean a borer is inside the tree?

I cut off the branch with the sap, and I saw no evidence of the borer in the wood. However, the bark on the trunk is peeling, and there some sap there too making me think it is root rot. I dug around the soil and saw almost nee feeder roots.

The borer would still be inside, maybe to hatch later; only cutting will tell if there is a hole in the center or a live grub. We have some problems with borers here; but mostly only when the lemons are planted next to coffee trees, which are hosts for the borer. Our workers are trained to see the damage; and to cut the branch below the damage and burn it.

I'll try to treat the borer (if it's possible) and further research the rotting roots I found too. Thanks for the input!

It could be gummosis. Borers leave a round hole. Gummosis damage is mostly slity.

I had that exact same thing on my yuzu this past winter! And everytime I'd clean it off it would ooze sap again and always from the same spot, since I didn't want to infect any other trees and knew that the yuzu was frost tolerant I put it outside in the snow and the problem resolved itself. I'm guessing the borer died in the freezing cold? Yuzu dropped leaves but it regrew them and is blossoming now! Not sure that helped but I wish you good luck :)