He has a 2-3yrold and you have been engaged for 2 yrs.
He knew she had mental health issues and left his baby with the mother to live with you . He is worried - so he should be - but then he probably should have been 2.5 yrs ago OP
You are not judging but your 3 yr old - is so much better. -OMG!
The child stayed over with you for the first time in over 2 yrs and she is less than 3yrs old, of course she cried.
Please, there is so much you are not telling in this post. You got together with a young babe yourself and he left his allegedly mentally unstable partner when the kid was less than 6 months old. If she is spending time with her Nan - so what, none of your business, at least she is being cared for. Sorry neither of you sound like you ahd the best interests of this child at the fore front of your minds and now you want to criticise!!!!
If he actually believes there is an issue then your DP needs to speak to the grandmother, get off his backside and sort it out. 2 yrs ago he left and he is only now having an ON....., speaks volumes really. Believe me looking after a child on your own is hard work, when you do not have psychological issues. I am as hard as nails but being a single parent has brought me to my knees on more than one occasion, exhaustion, stress, constantly being the one responsible, sole decision maker, no support, criticism from the OW, everyone giving me advice on how to parent my DCs = but not actually giving me a hand etc etc etc. You are in position nor is your DP until you have walked that walk.