How can I write a lens for a sum type

You are right in that you can write it with outside. To begin with, some definitions:

{-# LANGUAGE TemplateHaskell #-}

import Control.Lens

newtype Foo = Foo { _fooName :: String }
    deriving (Eq, Ord, Show)
makeLenses ''Foo

newtype Bar = Bar { _barName :: String }
    deriving (Eq, Ord, Show)
makeLenses ''Bar

newtype Baz = Baz { _bazName :: String }
    deriving (Eq, Ord, Show)
makeLenses ''Baz

data Problem =
    ProblemFoo Foo |
    ProblemBar Bar |
    ProblemBaz Baz
    deriving (Eq, Ord, Show)
makePrisms ''Problem

The above is just what you described in your question, except that I'm also making prisms for Problem.

The type of outside (specialised to functions, simple lenses, and simple prisms, for the sake of clarity) is:

outside :: Prism' s a -> Lens' (s -> r) (a -> r)

Given a prism for e.g. a case of a sum type, outside gives you a lens on functions from the sum type which targets the branch of the function that handles the case. Specifying all branches of the function amounts to handling all cases:

problemName :: Problem -> String
problemName = error "Unhandled case in problemName"
    & outside _ProblemFoo .~ view fooName
    & outside _ProblemBar .~ view barName
    & outside _ProblemBaz .~ view bazName

That is rather pretty, except for the need to throw in the error case due to the lack of a sensible default. The total library offers an alternative that improves on that and provides exhaustiveness checking along the way, as long as you are willing to contort your types a bit further:

{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE DeriveGeneric #-}

import Control.Lens
import GHC.Generics (Generic)
import Lens.Family.Total    

-- etc.

-- This is needed for total's exhaustiveness check.
data Problem_ a b c =
    ProblemFoo a |
    ProblemBar b |
    ProblemBaz c
    deriving (Generic, Eq, Ord, Show)
makePrisms ''Problem_

instance (Empty a, Empty b, Empty c) => Empty (Problem_ a b c)

type Problem = Problem_ Foo Bar Baz

problemName :: Problem -> String
problemName = _case
    & on _ProblemFoo (view fooName)
    & on _ProblemBar (view barName)
    & on _ProblemBaz (view bazName)

Sure, it's very mechanical:

problemName :: Lens' Problem String
problemName f = \case
    ProblemFoo foo -> ProblemFoo <$> fooName f foo
    ProblemBar bar -> ProblemBar <$> barName f bar
    ProblemBaz baz -> ProblemBaz <$> bazName f baz

It should be obvious how to extend this to further constructors, or even how to write a bit of TH for it provided you can think of a way to describe the right sub-lens to pick for each branch -- perhaps using a typeclass for dispatch or similar.

The function you probably want is

choosing :: Functor f => LensLike f s t a b -> LensLike f s' t' a b -> LensLike f (Either s s') (Either t t') a b

to be read as

choosing :: Lens' s   a      -> Lens' s'  a      -> Lens' (Either s s')    a

or in your case

choosing :: Lens' Foo String -> Lens' Bar String -> Lens' (Either Foo Bar) String

To use that with Problem, you'll need the fact that Problem is actually isomorphic to Either Foo Bar. Existance of both a Prism' Problem Foo and Prism' Problem Bar isn't sufficient for that, because you could also have

data Problem' = Problem'Foo Foo
              | Spoilsport
              | Problem'Bar Bar

I don't think there's any standard TH utility for giving such an isomorphism using more than one constructor, but you can write it yourself, which is somewhat easier than writing the lens onto the string yourself:

delegateProblem :: Iso' Problem (Either Foo Bar)
delegateProblem = iso p2e e2p
 where p2e (ProblemFoo foo) = Left foo
       p2e (ProblemBar bar) = Right bar
       e2p (Left foo) = ProblemFoo foo
       e2p (Right bar) = ProblemBar bar

and with that

problemName :: Lens' Problem String
problemName = delegateProblem . choosing fooName barName

Short version:

{-# LANGUAGE LambdaCase #-}
problemName = iso (\case ProblemFoo foo -> Left foo
                         ProblemBar bar -> Right bar)
                  (\case Left foo -> ProblemFoo foo
                         Right bar -> ProblemBar bar)
            . choosing fooName barName