I agree that trying to do your OW dives at 2 different places on a cruise is not the greatest idea for so many reasons, even without possible issues introduced by involving 2 agencies. I really think you're better off with the continuity of finishing up with one shop and one instructor, but aside from that your plans could be torpedoed just by being too congested to dive one of the days you planned, or by one of the places suddenly not having an instructor available at the time you need, or whatever - this stuff happens. Note the varied comments above on "if" and "how" and "can you" start with SSI and finish with a PADI shop - it's just this uncertainty and differences of interpretation and policies by different people and places that can make it a headache. Checking ahead is no guarantee of a smooth ride, as you can get "no problem" types answers from places in the Caribbean when you are a potential customer but find it's slightly more of problem once you're actually there. If you really want to try to complete the certification on a cruise through PADI shops, it might be well worth some extra driving from home to start it with a PADI shop and eliminate one possible problem.
(A friend of mine did SSI class and pool at home, then went on a trip someplace where her only option was PADI shops. She did in the end get her SSI certification, but it was a big hassle even after she had contacted the shop in advance to make sure things would work. In hindsight, she would have been better off completing a PADI certification there, though it would have involved more time - fine since she was staying there, not so fine if working within the constraints of a cruise.)
Another thing, if you don't intend to dive at home, when will be your next opportunity to go on a trip and dive? If it's reasonably soon, why not wait and plan a trip more centered around diving which is a much better way to do this? If it's going to be a long time, after doing just the certification dives and no other diving, you will not have the skills firmly ingrained and are likely to forget quite a bit of it and may find yourself mostly needing to start over. If that's the case I'd strongly consider a Discover dive of some sort - while it may not be the ideal way to go, it's practically made for this situation.