OK, here is how to get a really cool tsuba and also get it thinner.
Get some nail polish. I like "red she said", but you can use any color that looks good on you

Clean the brass well and don't touch it with your bare hands any more than posible.
Paint a border around the tsuba with the nail polish. It should cover the outer rim and extend about 3/16" to 1/4" inward on the front and back. This will end up as a raised rim.
Paint the center around and through the ana. Make it cover the surface about 3/8" around it. This will make the seat for the blade/habaki and tsuka.
Make up some FC about 5:1 ratio.
Soak the tsuba in the FC for fifteen minutes. Check the progress every five minutes and rotate the tsuba 90 degrees each check. It should start to eat away the exposed surface. Rotation prevents the bubbles from making deep grooves as the piece etches. Every 15 minutes, rinse well, check the polish for any that is lifting, touch up if needed, and continue to etch until the center is eaten away to a sufficient depth. At the 15 minute checks, give it a quick scrub with an old toothbrush to get the oxides off before putting back in the tank.
A micrometer is good for checking the thickness.
Once it looks eaten away enough, clean well, use acetone to remove the polish, and then clean up the surface. I start with a brass wire brush, then sand the high spots with 400 grit, and from there go wherever I want to. Bead blasting or sandblasting can look good.
If the look or depth isn't where you want yet, just re-mask the center and and rim and do it all again until the desired effect is attained. Sometimes, when it is close to the final look, I take the polish off the rim so it gets an antique look, too.
You can polish, patina dark and old, hammer distress with a tiny peening hammer ( this looks very good), or put on a matte finish. Hammer finish is done with an 8oz ball peen with the ball re-ground to a truncated cone with a small round end. This will leave tiny round dimples. It takes a couple thousand blows to do the whole surface, but it goes fast and is fun. Go around the surface in a continuous circular and wandering pattern ( don't concentrate on one area) and slowly fill in the whole surface. Do the rim separately, if needed.
Another way to get a cool tsuba is to do the above, but before you start, drill random holes from 1/8" to 3/8" throughout the surface to be etched. This will end up eating into slightly irregular holes and can look like an ancient tsuba. Use a sharpie to mark the places before you start. This avoids getting them too close or clustered.
The same thing can be done to make a zodiac tsuba. Mark and drill the star patterns for some or all of the zodiac or your favorite constellations ,,,, and etch away.
Other shapes can be sawn with a jewelers saw and then etched. Tree shapes are excellent.
The etching is slower, but the same technique can be dome with steel tsuba. Using aqua regia makes the etch faster.
The FC used for brass/copper will be no good for etching blades, but can put in a bottle and marked "COPPER" and kept for doing copper and brass. If you use it for steel it will auto-plate the copper on the blade and make weird colors.
The position in the etch tank will determine the look. If the tsuba is placed horizontal, the top and bottom will get a neat pebbly random look. I like this. It should be reversed top for bottom each 15 minute check.
Vertical hanging and regular rotation makes a different look. You can try several and see what you like.
It may be easier to etch the tsuba before the ana is cut. Just paint the area on the front ( or both sides) with a oval of nail polish sufficiently large to allow the ana and blade/habaki seat to stay unetched.