Hi and welcome!
That sounds like a good protocol for treatment, yes. Since it's a beta blocker, it's wise to monitor blood pressure when starting out.
Now, scarring: After the birthmark shrinks, there is usually minimal scarring. If a birthmark is surgically excised, that's where you have to worry about scarring.
I have known children who, because they were treated later in life, did have some unattractive stretched out skin left behind that had to be surgically treated. However, your son is so young, his skin has probably not lost its elasticity. I wouldn't think that scarring would be an issue for him, unless for some reason the propranolol didn't work.
The reason you can't get a straight answer out of anyone is because each birthmark is unique and reacts in its own way. In the cases that I know of, children who are treated early and proactively, like your doctor seems to be doing, don't end up with much, if any, sign that they ever had a birthmark.
My *opinion* is that you and your doctor are on the right path!
When will he go for his overnight visit?