What poor management skills your MiL has,
If you don't look after your staff, give support and execute a degree of flexibility, you find they don't have the same level of commitment to the company,
It's really short sighted to operate this way, if you do not retain staff, and the staff you do have, do not feel as dedicated to company as they should do, if they do not feel valued, the company can never be as good as it should be.
It costs a lot to advertise, interview, check references, and train someone new, very time consuming, energy far better spent else where, so the way the place you work is being run is very self-defeating.
If you have explained to management how important it is for you to have these couple of hours, and they have chosen to ignore your feeling on the matter, then you know how much the company values you.
how much time you have had for medical appointments has no relevance to this situation, completely ridiculous.
I would explain yet again (calmly) that you really feel that you should be accommodated on this occasion, if they are still not forth coming, make sure you put all your energies into finding a new job, preferable somewhere run with common-sense.
It is a milestone moment, so the outcome will jump out at you for years,
I would make sure that you are clear to MiL that the reality is, It will have a long term effect on relations, if she continues to go with the plan of stealing the moment from you and your child.