I would highly encourage the Peterson Method. It gives the artistic tools needed to make handwriting fun. It gives the students something to go on to really improve their cursive. CLP shows you the letters, but doesn’t really give the details needed to do it well. BUT Peterson needs to be taught in detail…holding pencils properly, muscle memory, etc. I have my teachers skip the cursive in CLP Language Arts and spend time each morning using the Peterson work books. There are lots of reproducible practice sheets from them.
We begin using this method the same time it is introduced in Language Arts, in second grade, but I think you could wait until 3rd grade. It is simple and fun (if the teacher makes it fun:)) and so young students can enjoy it. Peterson first teaches printing in Grades 1 and 2. Their 3rd level is where cursive begins. It has worked for us to use it starting with Level 3.