In your offer letter (or the like) it should give a point of contact to accept/reject the offer. That is where communication should be directed at the very least. I would consider including the professor in this communication regardless.
Now, let us be clear: Graduate school admissions are not a charity. This goes both ways. We should all get this. You were (rightly) clear that your acceptance was contingent on being funded. No one should be surprised that you are rejecting an acceptance offer.
I myself rejected a poorly funded offer for my MSci and then ended up attending the same university that I had rejected two years prior, this time for my PhD. (My funding was much better for the PhD offer!) No bridges had been burned.
As always, be respectful and be clear. Any reasonable professor/university will be completely fine with you rejecting an unfunded offer and then returning later to have a further conversation about PhD funding.