Registering yourself with the Embassy (or State Dept Website)
https://travelregistration.state.gov/ibrs/ui/ is mainly good for one thing....
If you should wake-up dead, after a night of debauchery, the local arm of Uncle Sam will have a head-start on identifying you, even without the passport. Additional, by updating your contact information, they will know who to contact so that the fees will be paid to transport your body.
Trust me on this one, through personal experience I know- they aren't going to send Clint Eastwood and the Marines to get your sorry ass from anything less than Medical School in Grenada (and that was
if they were already in the neighborhood).
if it makes you feel that you've dotted all of the i's, by all means, do it.
As far as the Policia running rampant on this Islands, I have trod upon soil of the Bay Islands for well over a year in-total, and not once did I ever have a bad interaction with the Popo. They are not likely to do much to help, but it is considered extremely unwise to mess with a tourist who didn't already need some messing with.
We are predisposed to view Central American (and foreign Law Enforcement) with a slanted view. Quite often, they are not constrained by laws in dealing with us as they might be in the US. This is because, indeed they do have different rules, and, surprise!
You are not a citizen. They are dressed more militaristically (it's cheaper), and on that point alone, a lot of NorteAmericanos get the willies.
A lot depends upon your attitude, any predisposition to interact poorly with uniformed authority, your blood alcohol level, or
whether you were driving. Be nice, smile, and move along.
In the US, the cops just don't want the paperwork. In Central America, the cops just don't want to be bothered. Move along.
The tourist to the Bay Islands interacting with the Police? Unless you go out of your way to invite that, you have 10,000x greater chances of seeing a Whale Shark.