Probably mostly legal rather than technical issues. Barging rockets down there would be an added expense over those of transiting to US based launch sites but if launches end up at a frequency that launch pad availability in the US becomes a bottleneck, doubt the additional cost would be much of an issue. And as you pointed out, for certain missions the added boost of an equatorial launch would be ideal.
Iirc Elon mentioned something about SpaceX being legally restricted from operating outside the US. Obviously there are national security implications, given how they're working on the most powerful rockets on the planet. My guess is it would unlikely be approved. Most of all they probably want the scientists and engineers in their own jurisdiction.
There is precedent. Rocket Lab is technically an American company yet launches out of New Zealand.
If SpaceX was selling a rocket to the EU or letting the Europeans operate the rocket a license might be difficult. But if it was only SpaceX personnel who had access to the rocket? I imagine a license would be likely. Not guaranteed, but likely.
