Those two ways aren't the only two ways for the characters to get trick weapons, nor are they equivalent; going either place will provide different degrees of freedom in which weapons are available and how willing the ones supplying the weapons are to part with them. Essentially each of the factions in the RP - the white and black Healing Churches, the Vilebloods, the Fire Dancers, the Followers, even the Harrow - along with the Hunter's Dream have the potential to be a source of proper Hunter-weapons, with more scattered across the IC world to be discovered and claimed, including quite a few original and unique ones. The Hunter's Dream is only known concretely to Nigel, and by inference to Morgraine, Torquil and Arcturus, making the White Church the only one actually known to them at the moment.

Do the characters have anything else they want to do at or near the Hunter's clinic, or are you okay with me fast-forwarding a bit, since at least Torquil and Morgraine appear to be opting out of interacting with the lantern? [@bloonewb] and [@DrabberRogue]?
(Also, maybe a little something just to give a gist of what Arcturus is thinking about all this?)