I am assuming the placing request school is full?
In my experience you have very little chance if that is the case. Schools round here are full to bursting, in the past you might have got in on a placing request to Bearsden Academy or Douglas Academy on a placing request but now all places are taken by catchment children so even a sibling is no guarantee.
All you can do is look at the criteria for admission and made sure they are aware of all of your circumstances. Usually siblings are fairly high up the pecking order when it comes to admissions, if there are fewer places than there are siblings requiring places, then it comes down to distance. Whether you live in the same local council area also makes a difference - I am assuming this applies in your case as you say your catchment school is geographically further.
What you can do is remain on the waiting list should a place come up, there is always a bit of movement in city schools especially between now and August. At least one of DD's friends who was initially refused a placing request had it granted by the time schools finished in June.
But this is always the risk of placing requests. To be brutal, the fact it's inconvenient for you to have two children at different schools makes zero difference to them and neither does the fact you don't drive. If your catchment is more than 3 miles away, transport will be provided for your child.
I wouldn't get your hopes up on professional letters either - when DD moved up there was a child in her year who had very challenging circumstances in that they had one disabled parent for whom they were acting as a carer, and the other parent had died at the Christmas of P7. The surviving parent had all manner of letters from the Head, GP, bereavement counsellors, all saying that this child needed to move on to S1 with their peers in a placing request school. Didn't get in after two appeals.
So long story short - check the admission criteria. Make sure you're on the waiting list. Prepare for not getting your way.