Good Morning Tyler, 

Thank you for contacting the U of M Extension Service. 

Interesting idea. Would you be injecting BT in the stem or the larve? How could you be sure the larve ingested the BT? 

Based on the information: https://extension.umn.edu/yard-and-garden-insects/squash-vine-borers#chemical-controls-3112162  I would start with a trap:

1. For trapping, use a container (e.g. pan, pail, bowl) colored yellow and filled with water, and a drop of dish-soap (break surface tension). Squash vine borer adults are attracted to yellow. They will fly to the container and be trapped when they fall into the water.
   • Place traps by late June, checking your traps at least once a day.

If I found the borer, I would follow the control suggestions: 

If your crop is still successfully attacked by borers, you can try to kill the borer inside the vine. Keep in mind that you may not be able to save the plant.

• As soon as wilting is noticed, use a sharp knife to cut a slit in the affected stem.
• Slice carefully up the vine until you locate the borer (or borers).

Once you have killed any borers with the tip of the knife, mound moist soil over the cut area and keep this spot well watered. New roots may grow along the cut stem, allowing the plant to survive.

The information states you need to kill the borers. It does not state you have to remove them. Although if I found one, I think I would remove it. 

You could surely try injecting BT with a syringe. I do not know what the BT inside the vine will do to your plant. That might be an issue. 

Good Luck!