Proving a specific language is regular
You can use the pumping lemma to show that a language is not regular. Here, you're trying to prove that if $L_1$ is regular then $L_2$ is regular. The pumping lemma can't be used to prove this implication. It could be used to prove the contraposite (i.e. you might be able to prove that if $L_2$ is not regular, then $L_1$ is not regular by the pumping lemma). However, even assuming this can work (I haven't checked), it would be a lot more complicated than necessary for this result.

You write “what if $v$ is non-regular?” But this doesn't make sense: the concept of regularity applies to a language, not a word. For a given $u$, what language does $v$ have to belong to, in order for $uv$ to be in $L_2$?

If that's not enough of a hint, try reasoning with regular expressions. If $L_1$ is regular, then there is a regular expression that characterizes it. How can you use this regular expression to characterize $L_2$?