None of them can be run 100% with a single bestiary
There is really no easy way to filter them per bestiary-availability (as we do to find adventures by level), so you must look at book by book and see if you have monsters that require another bestiary or you will be fine with the first.
Looking through the low-level modules I own, I found one module that has monster encounters that you can handle owning only the first bestiary book, though it will require adjustments on your pawns:
- Master of the Fallen Fortress (1st), you will be missing the Bat Swarm pawn though, and you can use a normal Skeleton pawn to represent a Burning Skeleton. There is also one encounter that will require 3 troglodyte pawns (only 2 in the box), but you can either replace one by another creature, or use another pawn as mockup;
Every other official module or adventure path book that I looked at will have at least one creature encounter from another bestiary book, or will have creatures that are unique to that book, so you will be missing her pawn (even though you do have the creature's artwork).
Even the first adventure in Rise of the Runelords, which is the most entry-friendly adventure path, will have some encounters that you will need a creature from Bestiary 2 like a few sinspawn (B2) encounters and a tentamort. There are also some goblin commandos that are from the book, so you won't have their pawns. But otherwise, pretty much the rest of the adventure could be run using only a single bestiary, though owning the adventure path pawns would also help with those bits missing.
However, keep in mind that I'm not including third-party adventures in my research, as I own very few of those and only one of them actually does fit these criteria, the first adventure in Trial of the Apprentice, The Bandit's Cave.
Also, it's worth keeping in mind that all creatures from all six bestiaries (and most of the rules contained in the core books, really) are open content. Meaning that you can find them online, and even print a personal copy to use in your games without any problem. Sites like the SRD and Archives of Nethys (which is now an official source) are good sources to find monster stat blocks.
Tuning down the requirements a bit...
If we reduce the requirements to something like 90%, or at least nine encounters out of ten being fully supported by bestiary 1 (B1), then this changes completely. As I mentioned earlier, the first adventure in Rise of Runelords could be run nearly entirely with a single bestiary. But there are several others that could fit into this too:
- Kingmaker: The first book has dozens of encounters that will only use B1, with a few encounters with creatures from that book, like the tatzlwyrm (later reprinted on B3), but otherwise no other bestiary is required;
- Carrion Crown: Again, the first adventure should be fine, with the exception of a single encounter with two crawling hands (B2);
- Giantslayer: The first adventure has an encounter with two gryph (B2);
- Ironfang Invasion: While the first adventure has lots of encounters with hobgoblins and their variants, there are only two encounters that will require another bestiary, a giant gecko (B3) and a shredskin (B4).
As you have noticed, many of them are designed to be mostly playable with the first bestiary, but will have one or another creature from the others.
It's worth pointing out that those I mentioned are only the static encounters in the adventure, as each book has a list of random encounters that are complementary if the GM feels like the group isn't getting enough XP/treasures (which may happen if they skip a few encounters) and they need additional encounters to fill that quota. Those random tables will usually pull creatures from all over the place, but are not required to be used in the adventures.
Disclaimer: There may be others that will fit into this criteria, but I do not own copies of all adventures and modules, but those I mentioned here are about 1/4 of my collection, so I hope it should be helpful enough.