On top of that, mothers are also expected to be in a constant state of bliss and fulfillment with their new roles.

If you said "fulfilment of their roles", it would mean the roles are fulfilled, not the mothers. The phrase is not common in that sort of context, but you may come across it in a commercial context, e.g. "fulfilment of your order is delayed because the product is out of stock". I'd probably use "by" rather than "with" here, because it is the role that fulfils the mother, if you think about the inversion of the statement. Again, you might come across "with" in an phrase like "fulfilment of your order with an alternative product". To answer your question about which word "with" is a preposition of, the answer is the noun phrase "their new roles". But I wonder if that is what you meant? Perhaps you mean to ask which word or phrase it attaches to? That answer isn't so simple. It could either be "fulfilment", or the noun phrase "a constant state of bliss and fulfillment" -- it is ambiguous. But if you intend the latter, then "in" would work better than "by", because "by" doesn't go well with "bliss", unlike "fulfilment", which is derived from a verb and would be used somewhat like a passive with "by".

I don't think "with" sounds good. There's no obvious preposition because of your construction "to be in a state of [noun]". The relation between the "bliss and fulfilment" and "their new roles" seems to be cause and effect / result. In this case, prepositional words and phrases like "because of ", "on account of" and "as a result of" would work. To make a very general connection, you need to add a word / phrase like regarding / in relation to / when it comes to. If you had chosen to use adjectives, this could work: "to be constantly fulfilled by and happy about their roles...."