Generating all sets might be computationally infeasible, but there's something very simple you can do:
Generate sets at random, then list all of the unique sums with 6 elements. Keep in a separate list all unique sums that have been found, and the set that generated them, and stop as soon as a unique sum has been found twice.
Listing the sums for a single set should be very fast, since 13 choose 6 is quite small (1716), and a random set generates on average about 17 unique sums, so two sets will collide with the same unique sum very quickly.
Unfortunately, this seems to generate one set with only small numbers and one set with only large ones. Maybe there's some possible tinkering with the parameters that makes finding two sets with a unique hard to find solution easier. My Python 2 code is here:
from random import sample
from itertools import combinations
set_sum = {}
un_sa = {}
def uni(s):
cnt = {}
for x in combinations(s, 6):
k = sum(x)
cnt[k] = cnt.get(k,0)+1
return [x for x in cnt if cnt[x]==1], cnt.keys()
while True:
s = sample(range(1, 51), 13)
u, alls = uni(s)
fs = frozenset(s)
fu = frozenset(alls)
mm = min(fu)
ma = max(fu)
set_sum[fs] = fu
for k in u:
if k in un_sa:
other = un_sa[k]
if len(set_sum[other] & fu) == 1:
print fs, other
print set_sum[other], fu
print set_sum[other] & fu
raise SystemExit
un_sa[k] = fs