This may or may not work - depending on the timing of the train from T4 to Chamartin. Although it's a short ride (about 11 mins), as far as I can recall, the trains are at 30 minute intervals. You could be lucky and jump on one when you arrive in T4 - or you might have just missed one!
The Tarjeta Dorada discount is probably about the same as the early booking (Promo) discount - so if you're determined to catch that train, you might want to book online to save time at the station. However, you still have to get your ticket for the local train to Chamartin (which is free when you show your Renfe booking) - so that means going to a machine or to the information desk.
As
@arthur1218 suggests, a taxi to Chamartin might be your best option if you're determined to catch the 10.15am train.
I have a similar dilemma for travelling to Leon in September. Our flight arrives at 9.55am and there's a train at 11.05 from Chamartin. I'm thinking of taking a chance and booking the tickets, even though it's a big risk. There's also a bus at around 12 noon, so that will be our Plan B - especially if I can bag one of the €5 promotional tickets!
Good luck and happy planning!