- prevent the edit button from being available on my own posts once I'm at the limit
This is not really a good idea from a user experience perspective. Stack Exchange's own Joel Spolsky wrote a blog post about this, arguing that unavailable comments should not be hidden or disabled, but rather inform the user why they're unavailable whenever an attempt is made to execute that command.
In this case, you don't want to completely remove the "edit" link because that 1) harms discoverability and 2) confuses people even more. But if the link is there and simply displays an informative message when one attempts to use it, we keep down confusion and limit the number of people posting "Eeek! Where did the 'edit' link go?" here on Meta.
The point is to educate the user on how to use the application. Hiding the option completely is utterly confusing, and disabling it doesn't do an adequate job of telling them why the command is unavailable.
- put the edit into the queue for review, regardless of my reputation
This is a workable option, and of your two suggestions, probably the one I'd have to vote for.
The disadvantage of this approach is that it requires the attention of two other users, who have to read your suggested edit, compare it to the original, decide whether it is acceptable, and accept/reject it. Worse, this takes away from the time and votes people have to dedicate to other suggested edits from people without editing privileges, and the queue is already pretty large at times.
Considering the purpose of this 5-edit-limit on your own posts is to prevent abuse:
This is an abuse-prevention mechanism used to stop users from maliciously destroying a lot of their content in a short period of time and denying users access to it. It's intended behavior.
We could probably figure out a way of fine-tuning the algorithm; i.e., by making it more granular. It does seem sensible to allow you to improve your own posts as much as you like.
Rather than preventing you from making any edits at all, we could simply prevent you from making more than 5 major edits (where some sizable percentage of the post's content is changed). This would allow you to fix typos and awkward wording multiple times per day, but still function as originally intended to prevent abuse/ragequit.
Alternatively, we could allow you to add as much as you wanted to your own posts, but limit you to removing content from posts only 5 times per day.