You should get accepted for finance so long as you have a good credit score - it shouldn't matter too much regarding that. You could also ask the garage how much they would take for your as a part ex toward the ST3 and re-negotiate the finance (if possible?).
Be aware, if the garage apply for your finance, that goes onto your credit history. So if you start shopping and getting various garages to put you forward to a finance company to see if they will accept you, over time (after a few of these) it will look bad - as on the history it wont show if you have been accepted or denied. Finance companies then look at this and see "Oh, hes applied for finance in the past week for 3 different loans" - they will then take it that these 3 have rejected you and will in turn reject you.
So, only go down the finance route once you are 100% that the price you see if the price your willing to pay etc (you probably know that, but working in a dealership, I knew a few people who said that they are waiting for 3/4 garages to get back to them to see if they can be accepted for finance and wanted us to put them forward to to another finance company, as they didn't think they would get accepted anyway) safe to say when I told them how it worked, they were a touch worried