There are some neat acoustic solutions (at least that would help) that wouldn't require you to drop your ceiling. Whether or not they're more expensive I couldn't say, depends on the details you ultimately decide upon.
For example, I personally love the look of wood slat paneling that can be used on walls or ceilings. Some options also have a thick black sound absorbing material behind the slats like felt. Some are rated for moist spaces like basements.
https://wallribbon.com/https://www.thewoodveneerhub.com/products/slatpanel-walnut-acoustic-wood-wall-panelsI've also seen creative acoustic ceiling felt tiles or panels suspended or integrated with lighting like:
https://arktura.com/wp-content/uploads/2017/01/soundedge-web-project-sandy-001.jpg