Hi Logan,
Theoretically, yes, you should be able to walk from the Hilton to the bus stop at
Disney Springs Marketplace and catch the complimentary bus to the
Beach Club Resort. We have stayed that hotel and used the Marketplace bus stop to get to dining reservations at resorts on the
Walt Disney World property with relative ease.
The distance isn't far, but you will have to cross a major road (which I believe now has an elevated walkover), wait for the bus, then walk from the bus stop at Beach Club to the lobby, all with bags in tow. My guess is that it would take you as long as an hour to do this.
For speed and ease, I would probably take a cab or Uber instead. Last week we were staying off property and had friends at the Beach Club. The Uber ride took just a few minutes and cost less than $8. In your case, I suspect that you could get from your hotel to the Beach Club lobby by taxi or Uber in about the same time it would take to walk from the Hilton to the Disney Springs bus stop.
Thanks for stopping by the Disney Parks Moms Panel with your question. We love helping guests with logistics and other tips to make your Walt Disney World vacation special. Enjoy!
Wilma