Have you cleaned it before trying to get all the tarnish off? Hot water with TSP, let it soak to get all the years of grunge off first. Then use the copper cleaner. This method works on silver too.

Yeah....don't try polishing dirt!!
Also Barkeepers Friend on the afore mentioned steel wool works wonders on really grungy stuff.

These were obviously stored in a damp basement. Yes, cleaning is a good idea. I must have ESP: just ran across my box of TSP down in the basement. (ESP for TSP?)

So�sequence: clean, scrub with BF and steel wool, then polish with brass cleaner?

Maybe this will finally cure me of picking up this stuff. Wish I could hire somebody to do this! I doubt anyone would have the motivation, though.

The really best way is to have a bench grinder with a buffing wheel....and some jewelers rough!....and someone to run it for you!!!!!

Twinkle and Barkeepers friend...
Also I have stripped chests with brasses that were also painted green and then red and then white, along with the chest. I found that the gallons of stripper I slathered on took not only the paint from the brass, but also the tarnish.
Actually it's likely that lye would work too....and not damage the brass.....but it does seem like over kill...unless you have something like a whole house with painted brass hinges and pulls.

Boil off the lacquer first, you cannot polish it away or polish through it, Then move on to ammonia and steel wool, Then if you want it very shiny, I have found that jeweler's rouge and a hard felt wheel is just the thing. A unique look is to go back with 0000 steel wool _after_ getting it mirror-finished with rouge, and "satining" it to the desired stage and grain.
Casey

Ammonia will remove the lacquer....certainly no need to boil.
And lacquer thinner will remove it too....lots easier than boiling.

what about baking soda with enough vinegar to make a paste and put it on and let it sit for awhile and it gets all the crude off, then polish. I just did that a few weeks back.