This is an interesting legal grey area that I have been sort of involved in with the Lego Group with a variety of communities that make films with Lego (brickfilms.com and bricksinmotion.com, among others). There has always been this yes/no relationship with the parent company.
Most people who have an opinion seem to be broadly divided into two main groups - those that say "the items have been paid for, the company has no say in how they are used" and the others say "the company has trademark/copyright on the item and can take action against an individual when condition X is met".
Usually "condition X" is defined as "starts making money" or "presents the product in an unsavoury light".
There was some controversy a while back when the film Frankenstein
was shown at BrickFest causing children watching to cry and calls from parents for the film to be banned (at least at conventions). Although there is nothing overtly violent in the film, the dark overtones along with Lego people getting knifed was enough to set the excitable off. Presumably the films shown before it were light comedies (the more common type of film made with Lego).
From the small amount of information available about this specific news item, the main problem appears to be the use of the company name, which is usually a bad idea. If the priest had used a more generic name, there might not have been as much of a problem. Or if the topic had been less hot-button it might have gone under the radar as well. Such things are usually done best with custom scratch-built models, but that is usually too expensive and difficult for the average person.
Plus it's usually something small and simple exploding that causes these big controversies. I can imagine what would happen if TLG told the other doctor to take down Legostar Galactica...