I also think the problem is likely to be mineral buildup inside the
faucet due to your having hard water.
I expect that CLR and Limeaway should be fine, but you might want to buy
a bottle of gelled phosphoric acid based toilet bowl cleaner for this
project at any place listed under "Janitorial Equipment & Supplies" in
your yellow pages phone book. You might also be able to buy the same in
the cleaning aisle of many home centers and hardware stores.
Phosphoric acid is the most commonly used active ingredient in bathroom
cleaners because it cuts through soap scum like a hot knife through
butter, but won't attack chrome plating; even at strong concentrations.
Being gelled, it will stick to the surface of the parts you're wanting
to clean without draining off, and that'll make the job easier because
you won't have to be continually appling more acid as it drains away.
And, like any acid, phosphoric acid will react aggressively with the
mineral deposits left by your hard water.
I'm sure CLR or Limeaway would still work, it's just that a gelled acid
meant for cleaning toilets will be thicker and would make the job easier
by not draining off the deposits you're wanting it to dissolve.
DO NOT use a gelled HYDROCHLORIC acid based toilet bowl cleaner on
anything that's chrome plated. Hydrochloric acid will attack the chrome
of chrome plated parts. Hydrochloric acid works well to remove marks
left by metal (especially aluminum) on glazed ceramic tiles or the
porcelain of toilet bowls. It's a good idea to have both kinds of acids
on hand for cleaning projects around your home. Be careful using
hydrochloric acid based toilet bowl cleaner. If you get it on your skin
it will start to irritate your skin after a few minutes. Keep a spray
bottle full of water handy to wash it out if you get it in your eyes.