Sorry to go on about this , but i just need to be clear here .
The joints are partially out of the holes - so i need to shove the joints fully back into the holes ( which i cant do because it's so tight )
So if you inject the product and it swells the joints , then it will surely be impossible to get the joints back into the holes ?
Tap the joints together with a mallet but put a piece of softwood on the part of the chair that you hit so that you don't hit the chair directly. You can hold the joints together with a 'spanish windlass' ie some cord and a stick.
For sure you need the joints together as they should be before any adhesive sets it into place. Glue them half out and you may as well sling the chair away. jomifl's advice is sound.