The letter you receive will be a systematic autoresponse that is part of their HR software. It's unlikely a human had anything to do beyond setting up the rule. At the end of the day there needs to be a line in the sand (deadline), and a policy of (appearing even giving consideration to late entries (it's happened to me before)) - that said, that sort of policy and rigid enforcement would be appropriate for a large org with lots of jobs advertised at any given point.
I'd be amazed if a polite call doesn't resolve this, and if it didn't, then that would be a whopping red flag that they are not only bureaucratic and impersonal but also idiots (which I suspect they are - midday deadline is a fucking stupid idea - end of (well, technically start of) should always be used). Operational (i.e. how you do things) stupidity or ineffectiveness is typically very cultural (i.e. the whole org is over-officious, or casual). If they can't accommodate this late entry, there will be a shitload of other crappy stuff that would drive me personally bonkers.
You did fuck up though, and need to own it, but the downside you will face is not worthy of the simple mistake. I'm not quite buying that it indicates you can't timekeep, but you have (more importantly, I'm afraid) can't read detail of instructions (which you are guilty of).
One thing I would say, is to think how you want to come across in the call. Write it down before you call. In your OP you sound pretty aggrieved (totally understandable, and as would I), and there's a fuckton of, essentially 'dog at my homework'. You will need to be absolutely contrite in your mistake (but NOT make a big deal if it), and the level detail may be offputting for someone, but remember that you are appealing to their better nature, and emphasise that you'd only do this due to how excited and positive you felt about the job.
Don't email - it will be ignored or batted back with an easier response to say no, and the personal touch is more appropriate for a request to flex the rules. Obvs don't go to the office, or stand outside with a sandwichboard for the lols
Do you have the HR contacts number? They aren't always publicly available due to recruitment consultants pestering them. Just think about that, and prepare what you need to say to the receptionist/switchboard if you have to go through that.
One last thing - HR people are generally not these days, office monsters dealing with rules and procedures (of the past) - there's a lot of softening (albeit commercially or legislatively driven) in their approach, especially in hiring. and, again, if they aren't then that's another bellwether of the type of place it is (full of dickheads).