However, a guildie did some further research and found that you can change those stats via the Mystic Forge the way you can change a regular (unnamed) piece. Thus, there's nothing preventing you from getting the weapon you want (thanks, Jaberse!).
However, as a reminder, the specialization weapons are:
shield x2 (mesmer, revenant)
staff x2 (thief, ranger)
These aren't the most universally-applicable weapons, and because of the overlap (assuming you only want one of each weapon), you can only get seven unique weapons.
The specialization achievements require you to level the character class to 80, fully unlock the advanced specialization, and then complete a bunch of expansion achievements/buy parts with expansion-area currency. However, once you have leveled the mastery in question to get the expansion achievements, you can easily unlock them for each level 80 alt you have.
My biggest problem right now is that I don't have all my alts at 80, and I don't have all the masteries unlocked that I need (currency will come just from getting the experience to unlock the masteries).
Therefore, knowing that you can change stats has not really changed my priorities. I already have an ascended greatsword and staff. I'd like to get the longbow and warhorn. I've decided that because my engineer is low on the list of alts, I'm going to just craft my guardian a hammer anyway, so after Ydalir/the North Wind, my main focus will be on getting a shield for my mesmer. Anything in addition will be for cosmetics/achievement hunting/character enjoyment more than minmaxing purposes.