The 2 big bike shops on Kauai is Brigg.s Bicycle (Lihue, near the port) and Kauai Cycles (Waipoli, part of Kapaa town), at least pre-pandemic. Check web site. Both are reasonably close to the Hilton. Don't know about the current rental status since the bikes shortage and tourist shutdown. Rental bikes were sold off by many shops because of the purchase demand & lack of rental use. Need to call and check.
Kauai is not really big into road bikes, mostly MTB.
Riding is not all doom and gloom on Kauai. If you're experienced at riding and use some caution it can be safe. Going north of the Hilton, it's not too bad, bike path through Kapaa and open highway up to Kilauea area (10 miles one way). South of Lihue is also open highway roads (30 miles). Ask about where to find a safe ride at the bike shop.