You’d certainly need the resources to be able to work unhindered, but I think it could probably be done with only a basic understanding of what’s necessary.
Let’s see if I can lash up a plan - this is without consulting anything but my own memory.
Prepare some clean(ish) working conditions.
Procure some growth medium - agar gel is the norm, but I’m betting filtered and resterilised stock made from animal bones would do.
If no petri dishes are available, jam jars (cleaned and sterilised by boiling)
Something to keep the cultures at a temperature somewhere around blood heat - a fridge (not functioning), a thermometer and a hot brick.
Now, some samples of pathogens - swab some nostrils and inoculate some clean cultures - wait to see what grows - expect to see colonies of bacteria - pick out some samples and isolate them on new clean gels. Streptococcus aureus is yellow - that’s what aureus means.
Moulds should not be hard to find. Sample and isolate some of these from rotting fruit, etc.
Set up permutations of cultures where a strain of bacteria is introduced on one side of the gel, a strain of mould on the other. Wait and see who wins.
Assuming a mould culture is eventually found that inhibits growth of bacteria, that one needs to be bulked up in volume - inoculate a large number of clean gels, wait.
Extracting the active ingredient is an unknown process to me - but I would try:
Pulverising with sterilised water, then filtering and boiling to re-sterilise
As above, with alcohol
Test the product on bacterial cultuires (could be a problem for the alcohol solution), then animals, then humans.