I don't understand why he doesn't understand

My DH and I get into these very strange arguments, which leave both of us feeling isolated and hurt. I feel like there are some basic things about relationships that he just doesn't get, and I try to be gentle about describing what I need, but he just doesn't seem to understand. Or maybe I am expecting too much? Maybe the problem is me? I think if I were better at figuring out how to talk to him we could resolve things better.

Here's a fairly typical conversation we had today. DH has recently started a job (about 2 months ago) after over a year and a half of looking for work. I have a job that I can, luckily, usually take DD (15 months) with me to. However, occasionally there are meetings that I can't take her to. She is mainly EBF, even at 15 months, and has never been away from me for longer than 2 hours. She is out of sorts when I am gone and generally cries until I get back. The only person she has ever been with is DH when I have had to go to a meeting.

There is a meeting that is 2 hours away driving time, then 3 hours at the meeting, then 2 hours back. I can't take her to this meeting. Its possible I might be able to get someone to go in my place, but I was exploring all the options first. I absolutely detest the idea of being away from my DD for so long. I know it will be horrible for her and I will be a nervous wreck. But if I have to be away from her then my first choice would be for her to be with DH. So our conversation this morning went something like this:

Me: "On Apr 3 I have to go to Seattle for the stupid meeting. Its mandatory. Do you think there is any way you could get that day off to be with Beantoes?"

DH: "Don't go to the stupid meeting. Get someone else to go."

Me: "Its my job, and I'm trying to think of how to do this."

DH: "I don't think I can take a day off. It would be inappropriate for me to ask. I am still on probation for another month."

Me: "What if you asked another employee if they could switch a Saturday with you?" (Since most employees would be happy to have a Saturday off)

DH: "Why are you dragging me into this? Why are you asking me first? You should ask E. or I. first."

Me: "Well, I really think its not fair to ask others to cover for me because I have a child. Child care is my responsibility. I would expect E. to attend the meetings she needs to attend even though she has to find child care, too. Why should it be different for me?"

DH: "*The Meeting* is a bunch of %&^$#. Just tell them you can't go."

Me (getting kind of agitated): "I really want to do my job. I'm lucky to have this job which allows me to bring Beantoes. I think I should attend this once a year meeting because its really my responsibility and I don't think its fair to dump it on E. or I."

DH (getting agitated): "Well I don't think its fair that you are making me talk about this stupid meeting on Sunday, my day off. I can't take a day off to watch Beantoes. I don't think its right that you are asking me this."

Me: "Okay, it sounds like you can't take a day off. That's okay. I just wanted to see if it was a possibility."

DH: "I don't think its right that this stupid meeting has caused us to be upset on Sunday! Talking about this is not what I planned to do with my day off."

Me: "I was just looking for support to figure out how to attend this meeting. I felt angry and unsupported when you said, 'don't drag me into this.' I was hoping we could work this out together, as a unit, not be working against each other. Remember when you had to do the CPR class and needed the car? I rearranged my work schedule so that I didn't need to go in that day, and figured out how to make it work so you could go to the CPR class. I was hoping we could talk about this meeting in the same way. Trying to figure it out together."

DH: "I don't understand why you don't understand that you can just not go to the meeting!"

Me: "I want to do a good job and do what's right and I was hoping you would help me and instead you're accusing me of ruining your day off and putting distance between us, and telling me that my concerns about this meeting aren't valid. And the meeting is mandatory; they say they won't renew our contract if someone doesn't go."

DH: "Okay, so you don't feel validated."

Me: "Or supported."

DH: "I would take a day off if Beantoes was sick or there was an emergency - but not so you can attend that stupid meeting."

Me: "okay, okay, I understand that its not going to work to have you take the day off. I would like to feel supported though in figuring this out."

DH: "Grrrrr just send someone else."

Me: "okay, okay."

I basically stop talking because he's just getting more agitated and think "I guess I'm going to have to figure this out myself; I'll be sure not to bring up stuff like this again since he probably can't help anyway."

Then there is this silence between us for the rest of the day. And sadness. I walk around on eggshells cause I know he is still upset and anything I say will just make it worse. I hate this. I come into the room and he leaves. We just don't seem to be able to talk about things. And this is just such a minor thing, really. I don't know how it got so blown up.

Any advice? I can take criticisms; maybe I was too sensitive?