There are certain rules your company should be aware of as you are a part-time worker you are still entitled to the same rights as a full-time employee.
Legally training should be scheduled as far as possible to allow part-time workers to participate.
Representatives are entitled to paid time for training as is reasonable to acquire the skills and information to represent their colleagues effectively and understand health and safety issues in the workplace.
Your training should either be scheduled during your shifts or, as you say, when your husband is not at work so he can look after your daughter. You have done the right thing by informing your supervisor of the situation and they should consider this when booking you in for training. At worst case scenario then your husband could take leave then you could attend then, but remember you are entitled to be paid to attend the training when it is outside of your working hours.
I would ask for a training time table for all the sessions you are going to have to attend, so you can plan this in advance.
You should not be forced to attend the training with your child, as I am sure this is against all health and safety rules. You should ask for alternative dates.
The Company could start a disciplinary against you, but I do feel this would be unfair as you have discussed your requirements with your supervisor and are doing everything you can to try and attend. It would be worth contacting the person who sent you the warning letter and discussing your situation further.