Collatz, Sort, Repeat
Python 2, 81 bytes

f=lambda k,n:k and sorted([x/2,x*3+1][x%2]for x in f(k-1,n*2))[:n]or range(1,n+1)

Try it online!
-19 bytes thanks to Leo and Bubbler
-11 bytes thanks to dingledooper (-6 + -5 from Python 2)
-8 bytes by swapping over to the brute force solution
-2 bytes thanks to Noodle9
To get \$s_k\$ up to \$n\$ elements, we only need to get \$s_{k-1}\$ up to \$2n\$ elements.