I have done exactly what you plan to do and it worked out with no problem.
A recruiter described a position to me and said he was going to submit my application materials. Weeks went by and I heard no feedback of any kind from the recruiter. After pinging him with many emails, his colleague finally replied and said, "Mr. X is away on vacation and cannot reply to your emails, but he's asked me to let you know that FooCorp has decided to pass on your application."
About 3 days later I saw a job posting on LinkedIn for the exact position that the recruiter had connected me to. I believed something fishy was going on with the recruiter, so I decided to apply via LinkedIn.
I applied and within two days the company responded to me and said they were very excited about my resume and wanted me to speak with their Chief Science Officer as soon as possible.
At that point, I mentioned to them that they may have already received a resume via the recruiter Mr. X.
The company thanked me a lot for telling them about this -- they said they would look into it and if they indeed had to honor any agreement with the recruiter then they would go ahead and do so. They seemed glad that I alerted them to this possibility.
I got no feedback about why the process went this way. That is, the recruiter (still, even after I began interviewing with the company) never got back in touch with me at all. And the company never confirmed nor denied whether they had received my resume at all from the recruiter.
In summary, it seems that if you have reason to feel you were not legitimately considered as an applicant when the recruiter was assisting you then you should definitely re-apply but try to communicate clearly that a recruiter may have submitted your information before.