You do need to push this back at your DH, if he wants to change his hours he has to sort out the childcare to cover that. He is a parent and he needs to negotiate with his employers about his work-life balance, it's not up to you to take all the hit. If it's not possible for HIM to sort out childcare HE has to decide if he wants to go on the training course or not, maybe that promotion might have to wait a few years, mothers make that decision all the time, why is it considered so terrible to suggest fathers should as well. Sounds like you've become the default parent to deal with this and it's really not fair that a) you've had to do all the phoning round and b) you have to take a week of annual leave for him to have a bit of a jolly. Being a parent does involve compromises both with work and your partner and your DH should not assume you will cover this, there needs to be a proper discussion about if you as a couple can manage this or not. And indeed the consequences of him going on this course, if he were to be promoted would he be working longer hours putting more pressure on you? Is that what you want as a family right now?
Practical suggestions:
How flexible is DH's employers? Can he work shorter hours on other days to make up for the long days on the days of the training course? Can he take the training course in another way, e.g. can it be run as a fortnight course if that is more practical for you? Can he do some of it as distance learning? There must be some flexibility in the course, if he was off sick on one of the days of the course they must have a way of covering that content, he needs to explore this with his employer. Does he know he can take parental leave, a block in the summer holidays could balance out any annual leave you have to take to cover this training course?
Is your work flexible at all? If your DH worked shorter days twice a month doing drop off and pick up would you be able to work longer hours those days to make up your hours? Do you work PT, can you change your days off to cover this or is your childcare not very flexible at the moment (we have this issue!)? Ditto parental leave, can you take it at a convenient time for you to balance out any leave you take for him to take the course?
Could the grandparents come to stay for a day or two to help out with some of the childcare, either for your DH's training course or during the school holidays?
The bottom line is that your DH can't make these decisions unilaterally, he needs to discuss both with his employer and with you if this is a) a good thing to do at all and b) possible to find a childcare solution for.