Starting BLW

My girl is going to be 6 months in 2 weeks and I’ve always planned to do BLW. Our pediatrician recommended we start cereal at 4 month appt but we hadn’t bc I felt like she wasn’t ready.

Fast forward we started giving her cereals at around 5.5 months but she hasn’t taken a liking to it so I wanted to take a little bit of a break. Pushing food out her mouth and when she actually swallowed it she threw up.

She sits well in her high chair but not really unsupported.

So the next phase for me is starting BLW in a few weeks so I’m wondering if she can’t sit necessarily unsupported, do I hold off?