The DM controls the narrative, not the mechanics
Random encounter during a long rest, with a watch, is a very common situation. What is the correct way to resolve it?
The "correct" way depends on the table expectations and the genre you are playing, not the game mechanics. To an extent, the DM can make sleeping in the wilderness as dangerous as she wants to. D&D 5e puts aside the mechanical approach to stealth, leaving it up to the DM almost completely. You can find some 'Sage Advice' on stealth in this podcast.
As a DM, think about decisions that the players could make and the consequences they should get as a result:
- Is (not) being on watch a no-brainer? What are cons and pros?
- Can players make meaningful decisions? Do they have or lack agency?
- Do they have fun?
The next paragraph is analysing the ambush from the players' perspective.
What was the point of the watch if it didn't prevent an ambush?
The very point of staying on watch is to be able to wake up the party before the danger bursts in. If the only advantage of the watch is the chance of the watcher (not) being Surprised, the watch itself becomes meaningless.
It's a good idea to give a chance of spotting the enemy beforehand. However, rolling for every bullywug and comparing with the passive score is the antipattern known as "rolling to failure". The rules suggest using a Group check instead.
To make a group ability check, everyone in the group makes the ability check. If at least half the group succeeds, the whole group succeeds. (PHB, Ch 7, Working Together)
It still requires a lot of checks for NPCs, which isn't considered a good practice. The game is a story about player characters, not about NPCs.
There is a better alternative.
Ask for a Wisdom (Perception) check from the player and compare it with the passive Dexterity (Stealth) score of the bullywugs. If he succeeds, let him spot the enemy far enough to be prepared. If he fails, the danger bursts in - all the teammates are asleep and unconscious. The watcher being (not) Surprised makes little difference in this case.