The BA lounge is airside, but, under the new security arrangements, it is before the security check, so you should be able to get to it on arrival. Of course, whether they let you in...
You don't want to be meeting anyone in Duty Free if you/they have lots of hand luggage - it's a busy rabbit warren and stores might not be open if SQ arrives at the crack of dawn. Best to stay safe, and meet past customs. NB There are several exits from customs - *A flights normally exit near door 34. Or you can meet at McD's/cafeteria in the basement (Level 2).