You don’t have to place a bet to get a drink at Bellagio. Just show up to the sportsbook, find a chair, and a cocktail waitress will be around to take your order. The Bellagio’s drink policy (they don’t even use tickets) is as lenient as you’ll find along the strip.
Las Vegas Hotel is another book that doesn’t go the drink ticket route. At LVH, you simply need to place a wager (the bet amount doesn’t matter), and then show the ticket to a cocktail waitress. The waitresses come around to both the general seating area and also the adjacent Fan Cave.
South Point began giving away freebies very liberally. You can likely get multiple drink tickets for minimum-sized wagers, in the neighborhood of $10 and $20. If you place a bet and ask for a ticket, there’s a good chance you’ll get one and possibly a couple.