Yeah it is definitely only a single shot per turn with the asp bow.
The asp bow in itself, is not a completely useless or terrible item; it does allow you to pick out characters in units and war machine crew and chariots etc. It does count as a poison attack, and characters/champions do not benefit from the 'look out sir' roll against it, and the fluff aspect of it is great.
What does it in for me is that it costs 30 pts.
If it was 25 pts, I'd more inclined to consider taking it. The reason for this, is innocence lost is THE no brainer bloodline power to always take. You really shouldn't leave home without it. It is the Lahmian's greatest strength. I've heard people argue before that with a Lahmian's high initiative, you can get away without having it ~ just avoid getting charged. Right. Well I for one do not want to risk the chance of being charged, and not being able to strike first. The way to keep your Lahmians alive, is to kill everything in base to base with it so they cannot direct any attacks back and take advantage of the lack of an AS. To do that, you need to strike first in all occasions (though if the rumoured HE changes about their special 'always strikes first' rule are true, then the Lahmians will be in even worse shape when it comes to facing up against HE, with it being a roll off each time). I also don't want to be having to be playing my game in a defensive/reactionary way, or in an overly aggressive way to ensure that I am not charged.
The point is, Innocence Lost, which imo is a must have, is 25 pts. Thralls have a 50 pt allowance. I can't take the asp bow, and have my thrall have innocence lost as well. So that doesn't work. I like to have my thrall have sword of battle, so it has that extra attack to help keep it alive. On my countess/lady, I take the attitude of that I need to take everything that I can to ensure her survivability. She will be my general after all, and if I lose her, my army is dust (pardon the pun

). So again Innocence Lost. She needs a ward save to make up somewhat for the lack of the AS, so I usually take Quickblood. I am intrigued by Draconis's kit though that involves taking the Crown of the Damed. I really don't want my general stupid, but a 4+ ward save would be even better. It is definitely something to try out I think and just see how it plays for me and goes. Then I like to give her sword of might to help make sure what she hits, stays dead. I take a magic item like a spell familar or the black periapt to help strengthen the magic phase, and help the army overall. With the left over pts, it's usually then Beguile. It won't work most of the time, and there are alot of ITP out there, but it is only 10 pts, and it is fluffy, and sometimes, it does work. Nothing more fun than having a champion/character trying to attack you and just them standing there with a doe eyed look on their face as you slaughter them
So again, spending 30 pts on the asp bow, I just don't have the pts there to spend on something that I don't think is really going to be able to do much. It still suffers all the normal shooting penalities except the -1 for shooting at a single mini, but then it has it's own -1 for it's special ability anyways. And she is going to be moving whilst she uses it (I am NOT having my thrall/countess/lady standing still somewhere just to avoid the -1 for moving), and if I do get lucky and get a hit, I then still have to wound, and then my opponent has to fail their AS if they have one. If I did get a wound against a character, you can be sure I wouldn't be given a second opportunity to get another one on that character, unless my opponent was an idiot. So basically, for the pts cost weighed up for what it can actually do and compared against the other options available, the asp bow just never gets a look in.