If I have a +1 shield and the Shield Master feat (PHB, p. 170), can I add 3 to my Dex saving throw when I use the second benefit of Shield Master?
If you aren’t incapacitated, you can add your shield’s AC bonus to any Dexterity saving throw you make against a spell or other harmful effect that targets only you.
Yes, it does
No distinction is made between the normal bonus to AC and the bonus as a result of the magic item. They are simply bonuses to AC. Since both are part of your "shield's AC bonus", they are both included.
Notice that neither of these bonuses are made distinct by a title like "shield bonus". They are just bonuses to AC that come from the shield.