If a process involves branding the forehead with magic, then lyrium is probably involved in some way, if not with the ritual itself, then with the process for creating the object. Of course since Tranquil are the people involved with making runes and magical items, it would seem that to obtain the necessary equipment requires the assistance of earlier tranquil. Therefore it is possible for Alric to by-pass the usual system, where he would have to go to say, Cullen, for the equipment, because all he has to do is to ask the Tranquil responsible to provide him with the necessary stuff. Origins showed us that Tranquil are a bit like computers, providing you supply the correct information, they will give you what you want - they do not question if you are entitled to have it. If Alric is senior enough to have signatory powers to gain items from the tranquil, then he will be given them. If not, then either forging a signature or coercing someone into giving him them would not be beyond his scope.
What I question more, is why no one queried the increase in the number of tranquil appearing (particularly if you believe Anders and they are obviously post harrowing mages living in the Circle). If, as he also states, it is illegal for post harrowed mages to be made tranquil, then Orsino at least should have been raising the roof about it. I posted elsewhere that I wonder if this is what led to the original breakdown in relations with Meredith, because Orsino was making complaints and she refused to look into it. You'd think that if a Templar had suggested mass tranquilisation of the entire mage population to both Meredith and the Divine, had been turned down, and then large numbers of mages start turning up tranquil, that it wouldn't be too hard to work out who was responsible.
Really it all comes back to the gap in Kirkwall between what should have been done according to the law governing Circles, and the reality for the mages who lived there.