You'd have to get very lucky to find a HOF game-used jersey for under $1K. But you can definitely find game-used bats, caps and batting gloves used by HOFers for under $1,000.
Just check all of the major auctions that run pretty much year-round. Grey Flannel just opened, and they have a ton of game-used items that may fit your criteria...
https://www.greyflannelauctions.com/catalog.aspx
Heritage, REA and SCP all tend to feature higher priced game-used memorabilia. But Hunt's, Huggins & Scott, LOTG, Crown, VSA, Mears and Grey Flannel feature high-end items as well as many more affordable game-used pieces. There is really no limit to what you might find.
Best of luck and Happy Collecting!
Ps. Awesome Fenway Table... Love it!