To be honest, as long as you're not obvious, like, as long as you're not obviously grunting/pushing/etc. when pooping, you should be fine to just go ahead and mess. Especially since you said you'd clean up right after.
As much as I wouldn't want *complete* bowel incontinence, I still say you should do what makes you happy.
And yes, while I'm not a mental health professional, I do believe you have some from of Body Integrity Dysphoria (BID).
I'm in communities for people with BID, and there are a lot of people there who do talk about wanting to be perceived as disabled, particularly in accordance to their specific desire, which varies based on the individual.
For example, a person with BID who desires to be mute would want to be perceived as disabled in the sense of being disabled by an inability to speak.
You seem to have BID and want to be perceived as disabled from being completely dual incontinent.