I really like your table top. I think it will finish out great. If you use a good wood glue, fit the pieces tightly and apply downward pressure, you will not need any clamps until you apply the borders. Your downward pressure does not need to be too heavy. Just enough weight to hold the piece firmly in place. If you cut some scraps about the same size as the piece your gluing in place, set the scrap on top and set some weight on it. Something like the weight of a gallon of paint. If you glue a few in place, you can remove the weight after 30 minutes and glue a few more.
If you have your project on the top of a table or workbench, you will be fine to glue up the entire table top without any worry of warping.
Good luck to you.
If you don't have time to do it right the first time, when will you have time to do it over?