I did some research. The strawberry tree in question is a Cudrania tricuspidata also called a Silkworm Thorn and native to China. Hardy to zone 7 and apparently to 5b with milder winters. Mine must be a male tree because it only produces a few fruit with no seeds. That's what one researcher said they'll do. The fruit did have a little taste after sitting in a window sill for a few days but still pretty tasteless.

Cinnamon. They won't cross over it and they will actively avoid it. It won't kill them but it will create a barrier. I've sprinkled it in my cherry and mango tree when they were actually in it and they fell off. I put a solid round of cinnamon sticks around a potted tree they'd been in and watched them run up near it and run away, trying again and again to find an opening.

Oh, San Jose? You should be fine in terms of chill hours. That oozing sap is very concerning though. This could be caused by many things. If you see any sawdust mixed in with the sap it could be caused by peach borers. It needs to be treated. This has probably been weakening and holding back the growth of your tree. If the oozing sap is black it is very likely Cytospora canker. There is no cure. You might consider taking out your peach tree and replanting a new one. And closely monitor the new one to preemptively and quickly treat any recurrence before it becomes too late. I would dig out all the surrounding soil too and replace it to reduce the chance of reinfection. There are ways to manage Cytospora canker if you really do not want to take out the tree, it involves taking out the dead and infected wood and continuously dipping the pruning knife in bleach to avoid spreading the fungus to the freshly cut open wood. But the base of the trunk is a difficult spot to try cutting into, likely may not survive cutting out the dead and diseased wood.

Looks tiny. Pot looks small too. Probably ripened too soon due to stress or it just didn't like life enough. Let us know how it was though. Flavor, sweetness, texture, if the fruit was fully formed inside. And some one info on the growth of the plant. How long you have had it. How long the fruit has been growing. Root rot or any other issues while the fruit was growing?