this question made me laugh because it hit so close to home.
During college I sometimes went over a month without showering ~:)
here are my tips wear clean clothes!
often times I find that it is actually my clothes that smell not me when you get home for the day, get naked.
the exposure to the air will dry off the sweat and stop it from "brewing" dust the stinky parts with baking soda, a great de-odorizer that shouldn't cause a reaction watch what you eat, eating a lot of processed foods and sugars can cause your sweat to be extra pungent wear gym shorts/lounge shorts under your pants, the extra layers stop the smell from leaking out sink bath, i know you said no bathing but just a quick splash of water and a stiff rub with a towel can help in a pinch witch hazel wipes for your booty, unlike wet wipes they should not cause a rash and help to stop any itching
you may be experiencing learn to wipe your butt after going number two, seriously if your butt smells like feces its because you are leaving feces there after you "wipe", wipe that stinky brown away!
lifehacks.stackexchange.com/users/1335/celeriko
Just take a shower.
Please.
Even if you are lazy about bathing, if you are experiencing the problems that you described in your question, it will take less time and effort to alleviate your olfactory issues by bathing relative to any other conceivable workaround.
lifehacks.stackexchange.com/users/7048/Yaakov Ellis
Good personal hygiene is important to keep healthy!
But you can put off shampooing your hair for a long time without it getting greasy by using dry shampoo.
lifehacks.stackexchange.com/users/7137/Silje Ljosland Bakke
Not to sound alarming, and I'm not trying to read into this too much, but you should know that it is considered a symptom of depression, or some other psychosis, when someone stops taking care of their own personal hygiene.
Feeling "lazy" about washing your body, or that this task is "too much work", are signs of depression.
It's not considered normal for someone to neglect their personal hygiene.
You may not feel that you are depressed, but are there other hygiene or personal care tasks that are also getting neglected?
It's a slippery slope.
Even asking here, about how to accommodate this behavior, shows some acknowledgement that this is not normal.
Cologne and deodorant don't count.
I'm not a doctor, so don't consider this as a diagnosis.
However, this is not normal behavior
and I hope you get motivated to either change your behavior or look into discussing how you feel about this with someone who is qualified to give you their professional opinion.
Web search on 'depression not bathing' .
lifehacks.stackexchange.com/users/9494/user70848
I will not accept Wet Wipes as an answer.
I find that Wet Wipes often give me a rash.
Try to find a wet wipe that does not contain Methylchloroisothiazolinone
lifehacks.stackexchange.com/users/10003/GisMofx
While I have never gone a week without bathing, I do find what I posted on another question works to control odor, which is to use surgical scrub.
A common brand of 4% chlorhexidine gluconate available at drug stores is Hibiclens.
(There is the veterinary brand Novalsan but that's 2% and doesn't work for some of the other uses I have for chlorhexidine.
I also use it in mouthwash.)
I've not had any adverse reaction and it also seems to help skin conditions which might be exacerbated by an irregular cleansing regimen.
lifehacks.stackexchange.com/users/6601/Nebuchadnezzar
You need to bath.
If you are really stinking bad then squeeze some lemon in the bathtub and add a few drops of rose water.
It will make you smell nice.
Also you can add liquid dettol(antiseptic liquid) to be hygienic.
If you don't want to bath then shave the hairs on your chest and underarm areas.
Hairs in these areas absorbs the sweat and smells really bad.
lifehacks.stackexchange.com/users/9602/Aashish Kumar Shaw
I'm a bit lazy about bathing, so sometimes I'll go a week without cleaning myself. I counteract BO by avoiding manual labor, so that I don't break a sweat; I also use deodorant and cologne. However, these prescriptions do not mask the smell of an unclean hiney.

How best to keep that area clean without bathing?

I will not accept Wet Wipes as an answer. I find that Wet Wipes often give me a rash.
How to Not Smell Like Feces without Bathing?
lifehacks.stackexchange.com
<cleaning><personal-care><smell>
lifehacks.stackexchange.com/questions/4252
lifehacks.stackexchange.com/users/2687/whitewings
To stop smelling without bathing, there are a few techniques that you can try. One of these is simply to wear clean clothes and shave your chest and underarm hair. You may want to also consider a surgical scrub. The options are to wear clean clothes, dust smelly parts with baking soda, avoid processed foods, take a shower, use dry shampoo, seek help for depression, find an irritant free wet wpie, take a bath with lemon and rose water or use a surgical scrub.
Wear clean clothes, dust your smelly parts with baking soda, avoid eating processed foods. Just take a shower - it's quicker than any workaround. Use dry shampoo. You may be depressed if not washing - try to find help rather than masking smells. Try a wet wipe that does not contain Methylchloroisothiazolinone. Use a surgical scrub such as Hibiclens. Take a bath with lemon and rose water in it, and antiseptic liquid for greater hygiene. Reduce smells by shaving armpits and chest.