Unless both her and her fiancée agree that they both want a baby .. Now . But if she were to wind up pregnant that will cause all kinds of hell and many questions about the how and why the current birth control didn't work . Yes I know none are 100% . But the relationship will have problems.

OP you just started a new job, put your energy into you job and performing well so down the road you will be eligible to take that time off ....You need to be employed for 12 months full time to qualify for FMLA. ( that is if you live in the US . So getting pregnant now is just not a good idea.

Have you considered being a foster for furry kids? Far to many are left in shelters when a foster home would be much better. All foster programs that I know of pay for food and vetting etc. This will give you something to help quiet the " I want a baby right now"
Loving and caring for a foster animal can be a mater of life or death for many animals. That would help you channel some love towards a furry friend. Trust me my dog often wakes me up in the middle of the night, lol

I do really do think that Therapy would help you. It often takes time to find the right Therapist.

Good luck