I'm no expert and I have no idea what you're swimming skills might be, but I think tethering yourself to your kayak is a bad idea. My son and I just spent a week on CC (Feb 13-18) and we dove on the ocean-side of the reef. We were in a 30' dive boat getting tossed around like a Coke bottle. I can only imagine how difficult a kayak would be to control, especially if you were tethered to it and snorkelling.
The reef is about 1 mile from shore, so you're going to be paddling quite a ways to get it. If you try to go to the ocean side of the reef, you're going to experience very rough, deep waters. The waters are very calm on the island-side of the reef, but they're also very busy with other boats.
I think "biting the bullet" and paying someone to take you is a better (safer) bet. We snorkelled the swash, the coral gardens and shark-ray alley for US$25 per person (that price included
the requisite parking fee). That is a common fee for the island tour companies. We were quoted US$22.50 per person by Mario's tours for the same tour, but we opted for French Angelfish Tours because of a time issue.
When you get to CC, talk to the various tour operators there. I'm sure they'll cater to your needs. French Angelfish adjusted their departure time for us, so I'm sure they'd adjust the stops, if you're not interested in the rays.
If you go to French Angelfish Tours, tell Marleni that Rahn sent you!! Also, ask if Heber can take you out.... they're a new business on Caye Caulker and they're great people!! Good luck and no matter what you decide to do, GO SLOW!