YANBU in asking your DH to spend part of one weekend caring for his children (for a few hours?) while you earn some extra cash which will presumably benefit the whole family or, at the very least, raise your self-esteem.
For the reasons given by ChippingIn, organise childcare and do the job as this will give both of you the powerful message that you are a woman of your word.
Endeavouring to communicate when the other party is non-responsive or monosyllabic can be a sure-fire way to raised blood pressure which may manifest in rage/shoutiing and issuing ill-thought out ultimatums etc. In this situation, tempers need to cool before any meaningful dialogue can take place.
Is your current frustration merely because of your desire to take advantage of a rare opportunity, or is it more deep seated - do you feel he doesn't pull his weight when he is at home? Do you feel that he is not attentive to your needs at any time?
Now that you've made your position clear and he's stomped off, I'd wait a few days to see if he contacts you but, no matter who contacts who first, say that you truly love and miss him (if you do), and that you appreciate how difficult it is for him to spend so much time away from his home and family even if you suspect he's got the easy option.
If soft soap doesn't make him putty in your hands , calmly say that it may be better if he doesn't come home this coming weekend as it could be beneficial for both of you to spend the time thinking about your relationship and future aspirations with a view to sitting down for a heart-to-heart the following weekend. NB: this ploy may hopefully achieve the desired effect, and leaves the way open for you to rescind your ultimatum while leaving your dignity intact.
Alternatively, and this would be my favoured option, don't contact him and wait to see if he grovels breaks first. In which case take full advantage.
If he doesn't make contact but arrives home at the usual time on Friday, act as if nothing has happened, take your dcs to wherever you've arranged childcare, go to work, and wait until your dh is back home full-time before you attempt to instigate a mutually meaningful dialogue about what you both want from your relationship.
As for tips on keepng cool when trying to communicate with someone who doesn't want to hear what you're saying, I feel your pain! All you can do is wear a pleasant face, engage brain before opening mouth, keep voice well-modulated, and don't rush your words. And take yourself off to another room/place if anger begins to rear its ugly head particularly if you're within earshot of your dcs.