You only need to get a new BRP card if you have changed your name in your passport, because the name on your BRP card must ALWAYS match the name in your passport.
So, you would only face a fine if you had changed your name in your passport and you DIDN'T change your name on the BRP card within 3 months of changing the passport.
The best time to change your name on your passport is when you have 3 months or less left on your current visa, because you will be applying for a new visa within those 3 months, which means you won't need to apply to change the name on your BRP. Instead, you just have the new visa BRP issued in the new name.
However, if you change your name on your passport when there's MORE than 3 months left on your visa, you will have to also change your name on the BRP too... which costs about £160 and can take several months.