Hi chocolatepeanut and may I first of all offer my sympathy. I am truly astonished that you have been unable to get a place for your dc at the local rc school. Is this your first child? I only ask because I am very surprised that you failed to put a second choice on your form but if this is your first child that is understandable as when it's your first you don't always know the right thing to do.
So I think you have many many things in your favour.
First of all, this person from the LED who advised you - frankly, I'd consider suing her. Or at the very least I would raise hell and tell the LED what advice you were given.
Secondly, I have never ever heard of a catholic child not getting a catholic education in all my 40 years of life. If there are more catholics then places, then the school needs to open more classrooms. After all, their sole purpose for existing is to educate catholic children - not say "sorry we're full".
Next, get the priest on your side.
Finally your dcs journey to school - if the school you have been awarded is 4 miles away then I thought (and I am willing to stand corrected here if anyone knows better than me) that the local education authority had to pay for and provide school transport. So if your father takes dc he should be able to take them by taxi and the LEA will pay for it.
The LEA cannot use the excuse that there are other schools nearer which your dc could attend because these schools are not catholic and you need a catholic school
I honestly can't emphasise how puzzled I am that you haven't been given a place. Do you live in an area with a large catholic community? Lots of Irish, Polish or Italians?
I expect what will happen is that if you send your dc to the allocated catholic school 4 miles away you will (provided you kick up enough fuss and get the priest on your side)soon be allocated a place at the nearest school.
On no account would I send dc to the other non-catholic school as the authorities might take this to mean that a catholic education isn't that important to you after all.
Can you say roughly whereabout you live?
Good luck and keep us informed