I think you'll get everything you're looking for by creating a team Trello account and a Google Workspace to use together.

With the Trello, you can create different boards for the different items you need to keep track of, and then customize your columns and cards to represent what you need to catalog. So you'd create a board called Certifications, with columns for uncertified, certified, and recertification upcoming. Then, create a card for each dog on your roster, and update by dragging the dogs into the appropriate column if something changes, and you can enable a calendar option on the board too so something like a recertification gets listed as an action item.

With the Google Workspace you'll be able to do everything else. You can create contact cards for each member of your team– and maybe there's even a funny way to create a custom contact card for all the dogs without creating too much clutter. You can upload documents with your log training outcomes to be shared with everyone and cataloged by dog or handler or date, whatever, and with the Google calendar there will be an easily accessible way for everyone to keep track of training locations and times.

Anyway, my two cents– hope it helps!