In my experience, it's usually better to do this prior to an application through a careers page, recruiters can often speed up or divert traditional process in your favor if things go well. This is not to say that there's no benefit to doing it after an application though, it just changes the approach a little. This is highly dependent on the companies size, but given you mentioned 3-5 internal recruiters with different titles and you're applying as an engineer, I'm going to assume it's a medium-large tech company.
If the position you applied for is within a specific team, it's going to be much more useful to try to find someone who recruits for that team, if you can't find them through LinkedIn, consider asking somebody who already works there and they might be able to guide you in the right direction. If it's a generic position and team selection is considered afterwards, I'd advise trying to find someone who best fits your criteria from the information you've seen on LinkedIn ("Graduate Technical Recruiter at Megacorp" / "Campus Recruiter at Megacorp" etc), if you're not sure, again, consider asking somebody who already works there.
In terms of messaging multiple recruiters, I think in a large company it's fine, but in a smaller company where recruiters might be working in the same office / desk, messaging every recruiter you can find could potentially come across as desperate or annoying, which you'd like to avoid. I'd personally keep any messages/requests simple, and try another recruiter if I don't hear back in a couple of days.