WOW. I was fuming over this till I got a chance to sit down and write. I've heard of husbands taking advantage of their wives, but this is off the charts.
I agree with Jennifer. You don't need your husband's permission to go!
Your husband is taking advantage of you, and he knows it. I can tell because he is talking out of both sides of his mouth according to what suits him:
1. He says basically that taking care of the baby all day is easy, and you get to take naps. BUT
2. He also says that he works hard all day and deserves to relax the whole rest of his daily (and nightly) life.
If taking care of the baby is so easy, he should be able to handle her while you sleep, do office work and everything else you want and need to do when he's home. Heck. He can even take naps.
I think another sign that he is deliberately taking advantage of you is, he refuses to go to therapy. My guess is, he knows the game is up the moment you two step into a therapist's office.
Also, what time is bed-time for your daughter? We used to put our daughter to sleep around 9-10. Whoops! Apparently many parents make it too late. We transitioned our daughter to 7-7:30 (8:00, some nights, just depending on how the day has gone and whether she's tired yet or not), and things overall are much better. This doesn't mean that she wakes up any earlier. In fact, babies sleep better and longer when they're not over-tired when going to sleep! Our daughter sleeps from 7-8 p.m. until 6:30-7:30 a.m.
Also, you can prolong those naps by intervening soon before her anticipated wake-up time. Go in there and be ready, and when she starts to rouse, cuddle her back, or do whatever it takes. It might take some experimenting, but you can do this! This will also get her better rested. Babies need at least an hour per nap. (Read No-Cry Sleep Solution by Elizabeth Pantly for more info - the book is Priceless!)
I say, take off and show him how much work it is, but not so that he knows how much work it is, but so that he knows that YOU know that he knows how much work it is. While you're gone, get yourself a lovely cup of tea and write up two columns. Write up everything you do every day, from start to finish. Then write up everything that you know your husband does at work (including that wonderful lunch break he gets - what is that, an hour? And does he get a 15-minute break or two, as well?) and at home.
Again, tell him you're doing this so that he knows that YOU know what the disparity is, here.
Put your foot down. Don't let him keep on taking advantage of you.
Hope this helps.