Five months is early days.
Replace the word ex with 'Mother' or 'Father'
How would you deal with a partner who had massive issues with a close family member that was difficult to manage due to 'boundary issues' on both sides or him having 'difficulty regulating his own feelings in regard to other challenging close relationships'?
Bottom line is that some people have more stuff to struggle with than others family wise and some are able to manage and compartmentalise it and others are still at the stage where they are learning to do that and others have not even begun to do that.
The question you need to ask yourself now is, 'Do I want to be part of this person's emotional struggles'?
Baggage comes in all sorts of shapes and sizes, it isn't just kids and exes, it is parents, siblings, work relationships etc.
If you're sense of unease is firing now at five months, due to his inability to manage his feelings down to a compartmentalised and 'you considering' place of neutrality after an emotional text from his child's Mother, then maybe you are asking yourself some VERY GOOD questions about the future of this relationship in terms of what you are willing/able to live with.
You can only deal with, what you can honestly deal with in life. That is not the same as putting up and shutting up or aiming for some sort of fantastic martyrdom born of love. Be honest FOR yourself and WITH yourself. Is this something that you can discuss with him reasonably? Is it reasonable to expect someone with other relational issues to compartmentalise them as they are happening or do that after a period of 'dealing' or would you expect that person to just not accept messages from family members who cause stress because of the way it impacts your relationship?
These are your questions and only you can answer them.
My best advice is that you can only ever really know your OWN best course of action and that your gut feelings of misgiving are very rarely wrong when thinking about how much you can take on, the too great sacrifice often means becoming resentful to the point that it explodes the world anyway.
Five months is very little to be already hacked off with ex and small child drama. Perhaps you would be happier in a relationship where neither of you had majorly differing baggage?