An option would be to bus to Disney Hollywood Studios and catch the boat ferry over to Beach Club. Probably less walking for the kitties and more fun for them to ride a boat over anyway
Also no park ticket required (or having to go through a purse/backpack check)
Exiting your bus stop at DHS, instead of heading straight to the park, walk towards the right where you will see large docking area. That is where the friendship boat picks up. Next stop Boardwalk, then Swan/Dolphin, Beach Club finally Epcot. At Epcot, it goes the reverse order. Entire ride GENERALLY takes about 20-25 minutes one way.
After dropping off kids, you can either walk or boat over to Epcot. You will be between England and France.
Going this way I would plan about an hour & half to travel to Beach Club - most likely it won't take that long

but this allows for missing the bus and the boat by a minute and having to wait 20 minutes for the next one
FYI, you can go to Fantasia Gardens WDW mini golf if you exit at the Swan/Dolphin, it is a short walk. It is a really cute disney themed course that is fun to do as a family, but it you'll need to allow even more time if you opt to do this.
That's what I like about WDW - everything is an adventure