travel prior to OCI application

I'm a UK citizen, my daughter was born in India two years ago and I'm still here, on a tourist visa which requires me to leave every 180 days. It's getting increasingly difficult to travel without her.
My partner is a citizen of India (from Ladakh, which under art. 370 has special status in India). We are not married (and do not plan to be). I have no interest or claim on any property here, nor do we have any Indian ancestry in our family. My daughter received her British passport last month. It currently has no stamps in it at all (she has not travelled anywhere since birth). We will apply for her OCI, but it's not done yet.
Can she travel? does she require an exit permit? if I take her out of India, can she return on a regular visa application? These questions are causing us a **lot of stress, would welcome guidance from anyone experienced / qualified in similar situations. Thanks.