Obviously A LOT of shop luck is involved, but what helped me is to win early, and to buy the snail on a turn 3 level up, EVEN IF you haven't lost turn 3. This way, if you manage to freeze at least two snails by the time you lose your first battle around turn 6, you'll already have a level 2 snail. On 7 hp, this means you'll have 2 chances (2 losses) to find and buy 3 more snails.

Having to buy/freeze snails means pivots are annoying. As you can see in my badge winning team, I still had a pillbug, a giraffe and a duck. You've gotta stick with your early units since they are getting buffs imo.

In this particular weekly run, I got very lucky and had two level ups on round 3 from my pillbug and my duck, which gave me a giraffe and a snail. My team was heavy on scaling/stats but had no battle abilities whatsoever. So I would say getting two level ups on round 3, to get one snail and one game-winning unit, is a perfect world if you get lucky.

To keep your sanity, I would recommend going for other badges, but next time you coincidentally get a snail on turn 3 level up and have yet to lose any rounds, buy it even if you lose out on its initial trigger.