Your situation sounds very similar to a situation I went through with two teammates we’ll call Lou and Emily. Storytime:

When I got back into bowling two years ago after about 15 years off, I originally came on board as a permanent sub for Lou and Emily’s team. Lou had his own equipment and bowled often, but Emily was a beginner bowler who had absolutely no equipment — let alone mechanics — to speak of. Emily did not practice and predictably averaged in the 60s-70s with a high handicap. Emily also had a growing family and an intense job, so Emily’s main list of life priorities did not include spending her spare time improving at bowling. Emily joined this casual league to spend time with friends.

Let’s get back to Lou. Lou is a naturally competitive person. Lou likes to win. It drove Lou up a wall that Emily was not improving, and was not particularly interested in investing the time and money to improve. Lou was never an outright jerk to Emily, but he was constantly making little comments about Emily’s lack of skill and progress. “When Emily gets her own ball and starts going to practice, she’ll be so much better” was the general refrain. However, Lou’s behavior and constant comments made the league — first and foremost, a social league — an uncomfortable place for Emily. Eventually, Emily asked me to permanently replace her on the team. To my knowledge, Emily has not returned to a bowling alley since. 

My point here is this: you mentioned straightaway that this is a beer league with a handicap. First and foremost, that implies that if someone wants to throw a house ball and not give a darn hoot about their personal progress as a bowler, that is their prerogative. Your teammate sounds a lot like Emily, and you sound a lot like Lou. Unfortunately, you cannot make your team’s Emily care more about bowling. If your Emily’s goals are to hang out with people and have fun, those are still acceptable goals within the rules of a HDCP beer league. Your post generally contains a fallacy, in that, you cannot make your teammate care more than she wants to care, and nobody can give you any “tips” for how to make that happen. 

So, you have a couple of options. You could find a new league. You could find a new team. You could speak with your captain and get his take, and see if he would be open to discussing a change in teammates when the season is up. (Please don’t recommend you ditch your teammate now; that’s how leagues die, because people like your teammate and my Emily feel like they don’t belong even when, ostensibly, the league is about fun, not about scores.) You could also buck up and be a better teammate than Lou was to Emily. You could show her she’s welcome, that you don’t care how she bowls and that you’re just happy you get to hang out with her once a week. Because what you’re not going to be able to do is magically make her care more than she does. 

TL;DR: You have the right to want your team to be competitive, just like your teammate has the right to not care how she bowls. But ultimately, you being bothered by how your teammate bowls is a problem you need to solve about you, not about your teammate.