OP, I would recommend you contact Social Services immediately and tell them you pushed your child due to overwork and stress, tell them you are scared that you did this and desperately need support. You should be assigned a social worker, who may actually provide some decent support, but more importantly it shows you took the appropriate action if your husband seeks to use this one lapse against you. Don't worry they will take her away from you - parents have done far worse, for far less, repeatedly, and never even had it suggested their child should be taken away. You will barely be a blip on their radar for this one impulsive mistake, but by seeking help for it you are keeping your nose clean and possibly accessing what support there may be to be had.
More importantly though, don't be a martyr to other people's expectations. It's very easy for them to say "Oh Marshmellow always copes so well" - as then they don't have to do anything. Don't allow that.
Be upfront with your mother, and ask for any help she can afford to give you, practical or financial. Failure doesn't come into it, you're drowning and you have to get whatever help you can for your daughter's sake. If it makes you feel any better, make it all about her and what she needs - food on the table, bills paid, some focussed time with a non-stressed adult. Ask for her.
And as for your &^(%ing husband... Sit him down and tell him you called SS about the pushing and are awaiting their feedback and a plan (that will knock that little catchall comeback on the head). Then make him listen as you outline the full scale of the problem. Then show him how the pay he would have had from returning to light duties would resolve it.
Tell him if he refuses to do so he is effectively condemning his own child to poverty or neglect, as you can't sustain this current workload and be a good mother. You are not a superhuman. He has responsibilities, that do not go away because he has an injury. You are not asking him to go back down the mines with a broken back; just to take the generous offer to return to work he can manage.
Tell him if he will not do this, he clearly does not care for either you or his child, you can no longer love and support someone like that and you want him to move out as he is not contributing any practical or emotional support and is dragging you down with him.
Frankly I'd chuck him anyway, he sounds like a vindictive manchild.
Seriously though, the world is shitting on you right now - do not draw any more shit to yourself, you can't afford it just for the ego-boost of being the one everyone relies on and praises for 'coping'. Let go of that role, embrace your human side and your limitations, be upfront about them and protect yourself - and most importantly by doing this protect your daughter. If you carry on as you are, you will break again and she may well be in the line of fire.