In period the type of wood they used was usually not a, "You're making an X? Oh, then you'll using Y." kind of thing. Evidence tends to point to their using species based on their properties (hardness, rot resistance, likelyhood to split) and their availability. (It is, aparently, not uncommon to find pieces with a mix of species, each chosen for their particular strengths.)
Looking at the picture, but not really knowing much about what stresses are put on such a loom or about the particulars of any of its componens, I'd say just about anything they would have had access to would be acceptable. The wood used in any of the boxes recovered from the wreck of the Mary Rose would probably be acceptable. This, of course, depends on your kingdom's A&S judging system and how fussy the judges you draw want to be.
I don't know the exact date of the tapestry or the location you are aiming for, bubut dovetails were fairly rare for most of the SCA's period. Nails might be acceptable, but given that it sounds like it's going to spend a lot of time in and out of your lap, you might regret that decision as well. I'd not go with the butted ends for reasons of durability.
Personally, I'd recomend tongue and groove joinery for this.
Are you planning on attending Gulf Wars? I know a number of us will be there with hand tools.