The authentic colors for an 1810 bathroom would be whatever the chamberpot was glazed with. Don't worry about carrying colors over ... treat the bath like an 1890s upgrade back when great-great-granny told her husband that either they got indoor plumbing or she was going home to momma and he could have the chamberpot. F&B's Arsenic might work, as might St Giles Blue and Cook's blue. There's also the classic pastel paint - pale grey, pale yellow, or pale blue

You are right on track for making those changes to the space. The lighting upgrade would be a great start! Lightening up the cabinet color would make the space much brighter. Since you have a lot of dark colors and patterns in the bathroom, a white cabinet would freshen up the space. Best of luck!

The last thing I consider especially for exteriors is which direction the front door/windows faces. Because the light is constantly changing throughout the day and the year - it's a point of criteria that literally changes by the second. I understand your attraction to the cleaner whites in terms of chips and samples. What you need to try to anticipate is what happens to white on a bigger scale and in sunlight. Just because the house faces north, doesn't mean it's on the dark side of the moon. Sunlight - from all directions - strips chromaticity or colorfulness. So that "clean" white that looks good on samples and chips ends up looking rather arctic. Simply White is a really good suggestion because it has a little bit of chroma/colorfulness and there's a chance it will "wash out" and look just right once installed. As a general rule, you can always go a little darker and a little more colorful on exteriors than you think you can because, again, sunlight will knock back the intensity.

I live in a house that was built in 1926 and the bathrooms were remodeled about 55. I have since re-remodeled...but that blue tiled bath made me very nostalgic! The toilet and tub are most certainly original as is the wall tile. However the vanity and is newer and perhaps that large mirror. But love the set in toothbrush holders and the soap dish with the grab bar.....and I had towel bars exactly like you have!! Mine was sandle wood and dark brown....and the wall paper was tiny flowers....quite subdued. If you want to stay true to the period, I would do wall paper.....and a small print. for the one with the yellow floor, I suspect the toilet in new, It likely was yellow, ans of course the vanity....but the tile is very of the period. I would paint the walls about the same color as the floor and accessorize with tropical green.