What is a good technique for binding a saddle stitch booklet with a hard-ish cover?
When using 120 grams paper, it's best to use several signatures/booklets of 4 pages each. Then you can sew these together using a bookbinding technique (I like coptic binding - YouTube has great tutorials). 

If you don't want to use a different bookbinding technique, you can make your booklet flatter by folding all the folded pieces separately and making sure you have a sharp fold before putting everything together.