What options are there so my nose doesn't move when smiling after rhinoplasty?

this problem is mainly a result of poor tip support - you would likely benefit from structural grafting of the columella and tip

In some patients, the attachment of the nasal muscles in the central portion of the nose can pull the nose down while smiling. The easiest way to correct that is to snip that muscle during a rhinoplasty procedure and place a graft to prevent it from reattaching and support the tip of the nose. I hope this information helps.

Nose droop on smiling is a common problem when the tip is ptotic and underprojected. It can also occur when there is "tension' in the nose because the nasal septum is pushing it down. rhinoplasty can correct this for you permanently.

This not an uncommon issue. Usually I cut the upper lip muscles slighly and most importantly, I support the tip with a cartilage intercrural graft. Cinching sutures have been used to narrow the nasal base but, often lead to asymetry and discomfort.

By the sounds of it, you have a plunging tip.  You had this addresses and want it back.  There are a number of surgical maneuvers which can be done to address this issue either by way of muscle cutting technique, implant or grafts.

 Photos are always helpful but from what you describe the nasal muscles overcome the inherent strength and support of the nasal tip, when you smile, causing the tip to droop.  The cartilage at the upper, anterior end of the nasal septum can be trimmed and an orthopedic permanent suture placed to rotate the tip and add strength to resisit the muscles effects. 

 This is a fairly common procedure, have done this many times and can be done with or without Rhinoplasty depending on what else is done to the nose at the same time.  By itself it's a 15 minute procedure with incisions in the inside of the columella.

When your tip pulls down, it is not because of your upper lip or lip muscles; that's why the lip surgery didn't work.

The pulling down  is because your nasal tip cartilages aren't strong enough to hold your tip up and resist the pull.  This is called "inadequate tip projection" and is very correctable. 

Find an experienced surgeon whom you like and see how he or she an help support your tip better--my preference (because they are anatomical) is tip grafts.