LOOKING FOR BUCK-BOOST

See if you can split your output into 2 pieces and use 2x TPS63020.

I do not think it will worke for the customer.

Is the TPS63010/1 are for  for 3A max output?

I'm not sure that I understand the question about the TPS63010.  The D/S for each device has a figure that shows output current versus Vin.  This is what you need to look at.

The TPS63020 offers the highest output current, among the TPS63xxx family.  The integrated nature of the TPS63xxx family makes it easy to design.