My son is the same age as your boy, and I help coach his little league team. They are a very good team, and a number of the kids are very good players. I am the third coach on this team; that is, our league allows teams to protect the manager's kid and one coach's kid, and I am not either of those coaches. I managed my son's machine-pitch team last year, but like to assistant-coach for a year in a new league if I can swing it. As a result, I am the dugout coach, or what I like to call the Mommy Coach.
We have poor players on our team. For example, one kid, bless his heart, is league age 11 (one of two 11 year olds in the league). He has an almost pathological fear of being hit with the ball and is rarely in the batters' box when the ball gets there. As a coach, I will not put up with comments that are overly discouraging. My strategy is usually to take the two or three best players aside sometime, emphasize their leadership role, and that they are setting an example for the rest of their teammatges. I tell them they need to step up in leaders by encouraging their teammates, not discouraging them. This age kid will respond to that. "We need everybody on this team to contribute, and x kid needs to be encouraged to contribute, etc."
Maybe you need to find the Mommy Coach on your team, rather than looking to the manager. It is hard for some coaches to deal with this issue, because the snide kids are, in fact, also only 9 years old, and really can't be expected to be as mature as grownups. Also the coaches often have a lot of other stuff to deal with and don't see everything, and they almost surely don't see the worst stuff, which is usually outside their immediate presence, whereas you are trained on your kid all the time. Kids at this age, even the best players, have their own insecurities and issues, a lot of which come out when there is an easy target. This is true of one kid on my team who I believe is the best player in the league. We want those kids to understand their role and be big enough to encourage rather than discourage, and a lot of time, they're not gonna get there on their own without help.
So my first suggestion is to see if there is a Mommy Coach who is locked in to the dynamics of the team. Sometimes a well-placed comment or two from this person will reduce the ribbing.
The other strategy might be to talk to the manager. Don't think of this as complaining to the manager. Hit him where he lives. Say, look, I know little Jimmy isn't the best player, and I'd like help him improve. Do you have any suggestions for things I can work on with him? I was thinking of ... (fill in any number of the fine suggestions given above, like working through situations on a piece of paper, maybe a video game - Backyard Baseball is fun, etc.). Oh, and also, little Jimmy really gets down on himself when his teammates get on him for mistakes. I think he might be able to focus better if he got a little more encouragement from his peers.
Sounds like this isn't true, but who cares. This is the language the manager speaks. He can sell "we are a team, we need to play like a team and encourage each other so we can all be our best." He can't sell "Don't be mean to the dorky kid."
I don't see any reason to place a label on the kid as explanation for his play. It's a manager, not an MD. It is no secret that there is some issue, whether it is "just" a coordination problem or "just" a focus problem, or those problems because of some underlying condition, doesn't matter at all to the manager. If he cares at all, it is about how to solve the problem (hence your request for how you can help little Jimmy at home). It's not like he's gonna say, oh, an Asperger's kid, why didn't you say so before; I know just how to change my approach to work with him.
If this doesn't work and the negativity can't be controlled, I think those are not the right kids for little Jimmy to be around. Doesn't matter if little Jimmy wants to or not; you are the parent and you gotta call 'em like you see 'em. You have already made this call on schooling. I agree best case is for it to work out, but if it doesn't work out, Plan B is to find better kids. Maybe you can play out of district, or talk to other parents about other managers with a better approach. That might change everything. Manager's kid is usually the best player, or one of them; while the manager's coaching matters, his parenting matters more in this respect - if he raised his kid to be a good kid and his kid encourages instead of discourages, the other players will fall in line.
The sociological dynamics of a little league team are interesting and finely balanced. A single comment from a high-prestige kid either protecting or attacking your kid could either insulate that kid for the rest of the year or open him up as fair game. There is an interesting book called With the Boys, written by I think Gary Alan Fine, about the sociology of the little league dugout. It is a little dated and with Major level kids, but still an interesting read. Maybe someone has updated work on the topic since, don't know.
Best of luck.