Hello @vasilis4sar
Thanks for clarifying what your client requires.
As my colleague explained before, Facebook for WooCommerce doesn’t allow any specific option to exclude the price from syncing.
You can, however, show a different price between WooCommerce and Facebook in the catalogue:
https://d.pr/wS4oks
This does not removes the price from the product either.
I’m afraid what you are looking to do is not possible with the Facebook integration. You can check with Facebook support directly to know about options on creating Catalogues without price, as far as I know there is no way of setting this up.
You can check their documentation here: https://www.facebook.com/business/help and consult with a developer to know if this can be excluded somehow.
What you are looking to do is similar to what this extension does: https://woocommerce.com/products/advanced-catalog-mode-for-woocommerce/ but there is no integration between that and Facebook. This will only work on the WooCommerce shop.
Additionally you can include this to our ideas board here:
https://ideas.woocommerce.com/ where developers go to look for future plugin features and improvements.