British Passport or Indian Passport with RoE certificate

Hi All

I am an Indian national in the UK who currently holds an ILR. My baby is born after my ILR and I understand that my newborn is eligible to directly apply for a British Passport. I have a few queries in this regard

1. If i apply for my baby's british passport, which card (PIO or OCI) i should apply for? I plan to travel with her in the next 1-2 months. How long does OCI and PIO take to come?

2. I am keen for her to have an Indian passport as well. I understand that if i get a british passport, i wil not be able to get an indian passport. The alternative is to 1) register her as an indian citizen 2) apply for her passport 3)apply for right of adobe certificate from ukba. Is this approach correct ?

3. If i follow the above approach, would it be possible for me apply for her british passport AFTER getting an indian passport? Would it pose any problem with her getting a British Passport later (as she is by default a british citizen)

I would be keen to know if anyone else has had the same situation and how did they go about sorting these issues?

thanks