My husband came into our relationship *knowing* that I had a 1 1/2 year old (at the time). He was great with her - and she adored him - but now that we've had our second daughter he just seemed at a loss. I think that I got to have a unique perspective on this problem by seeing my husband's flip. Suddenly, the man who had always taken MY daughter out to play at the park so I could have a relaxing bath was CONSTANTLY handing me the baby, refusing to do anything unless directly asked, and generally snippy with me about the whole thing (not to mention the "is she hungry?" fights we've had!)
We've talked about it and came to the realization that what was actually happening was that my husband was scared out of his skull - and taking it out on me, since he couldn't/wouldn't take it out on the girls. He felt useless - I was the only one who could feed her, I seemed to have all the answers, so why wouldn't I just DEAL WITH HER?
Perhaps your SO has the same problem? Maybe all this tension is really his way of coping with the fact that he doesn't know what to do with your children? Feeling lost and helpless isn't something that guys tend to know how to deal with well - and, as I'm sure we can all relate!, children can make you feel the MOST lost you've ever felt in your life!
Perhaps talk to him about this. Tell him how hard it is on you to deal with the kids too - by letting him know that YOU don't have it all together, perhaps it will open the way for him to admit the same thing (if only to himself!). Then perhaps you could talk about some things that he has trouble doing for the kids and the solutions, tricks, or tips you've devised for dealing with that same issue. Don't forget to ask him the same thing! He may have figured out something you haven't, after all ;)
Another thing - you mention that he resents you going out and leaving him with the kids to go to the gym. Does he have a night out? If not, perhaps you could suggest it to him - something to do besides go to work and deal with the kids. One thing that has also helped my husband and I deal is that we have friends over at least once a week. Most of them don't have kids, so we do an after-bed-time gaming session where we play board games, card games, or RPGs. Perhaps you could do something similar, or even just a TV or movie night to watch something you BOTH enjoy with friends that are either friends you have as a couple or a mixture of his friends/your friends. Having some time to relax with grownups is necessary to our mental health! Life isn't all Blue's Clues and cold chicken nuggets - and we can all usually use a reminder of this!
In the end, try talking about it together when neither of you are upset. Agree before hand that if the conversation becomes heated you'll break off for a 10 minute break to calm down - and do your best to keep your heads! If all else fails, counseling is a good route to take (although it does cost money - which can lead to more problems in some relationships!). In the end, if you find you just can't make it work as a couple, perhaps it *would* be best for you to break it off - but that doesn't mean he isn't their father all of a sudden. Plenty of families function better with joint custody than a joint house - it all comes down to what makes you and your SO happiest. Remember, if you aren't happy, your children won't be either.